
Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th
11th Edition
ISBN: 9781305965737
Author: Dennis G. Zill
Publisher: Brooks Cole
expand_more
expand_more
format_list_bulleted
Concept explainers
Textbook Question
Chapter 2.4, Problem 13E
In Problems 1–20 determine whether the given differential equation is exact. If it is exact, solve it.
13.
Expert Solution & Answer

Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
This year Amanda decides to invest in two different no-load mutual funds: the G Fund or the L Mutual Fund. At the
end of each year, she liquidates her holdings, takes her profits, and then reinvests. The yearly profits of the mutual funds
depend on where the market stood at the end of the preceding year. Recently the market has been oscillating around
level 2 from one year end to the next, according to the probabilities given in the following transition matrix :
L1
L2
L3
L1
0.2 0.4 0.4
L2
0.1 0.4 0.5
L3
0.3 0.3 0.4
Each year
that the market moves up (down) 1 level, the G Fund has profits (losses) of $20k, while the L Fund has
profits (losses) of $10k. If the market moves up (down) 2 level in a year, the G Fund has profits (losses) of $50k, while the
L Fund has profits (losses) of only $20k. If the market does not change, there is no profit or loss for either fund. Amanda
wishes to determine her optimal investment policy in order to maximize her (long-run) expected average profit per…
Evaluate
F. dr where F(x, y, z) = (2yz cos(xyz), 2xzcos(xyz), 2xy cos(xyz)) and C is the line
π 1
1
segment starting at the point (8,
'
and ending at the point (3,
2
3'6
Solve this questions please
Chapter 2 Solutions
Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th
Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - dydx=1xy (a) y(0) = 0 (b) y(1) = 0 (c) y(2) = 2...Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - In Problems 512 use computer software to obtain a...Ch. 2.1 - In Problems 512 use computer software to obtain a...Ch. 2.1 - In Problems 512 use computer software to obtain a...Ch. 2.1 - In Problems 512 use computer software to obtain a...Ch. 2.1 - In Problems 512 use computer software to obtain a...Ch. 2.1 - In Problems 512 use computer software to obtain a...
Ch. 2.1 - In Problems 512 use computer software to obtain a...Ch. 2.1 - In Problems 512 use computer software to obtain a...Ch. 2.1 - In Problems 13 and 14 the given figure represents...Ch. 2.1 - In Problems 13 and 14 the given figure represents...Ch. 2.1 - In parts (a) and (b) sketch isoclines f(x, y) = c...Ch. 2.1 - (a) Consider the direction field of the...Ch. 2.1 - Consider the autonomous first-order differential...Ch. 2.1 - Consider the autonomous first-order differential...Ch. 2.1 - In Problems 21-28 find the critical points and...Ch. 2.1 - In Problems 21-28 find the critical points and...Ch. 2.1 - In Problems 21-28 find the critical points and...Ch. 2.1 - In Problems 21-28 find the critical points and...Ch. 2.1 - In Problems 21-28 find the critical points and...Ch. 2.1 - In Problems 21-28 find the critical points and...Ch. 2.1 - In Problems 21-28 find the critical points and...Ch. 2.1 - Prob. 28ECh. 2.1 - In Problems 29 and 30 consider the autonomous...Ch. 2.1 - In Problems 29 and 30 consider the autonomous...Ch. 2.1 - Consider the autonomous DE dy/dx = (2/)y sin y...Ch. 2.1 - A critical point c of an autonomous first-order DE...Ch. 2.1 - Suppose that y(x) is a nonconstant solution of the...Ch. 2.1 - Prob. 34ECh. 2.1 - Using the autonomous equation (2), discuss how it...Ch. 2.1 - Prob. 36ECh. 2.1 - Suppose the autonomous DE in (2) has no critical...Ch. 2.1 - Population Model The differential equation in...Ch. 2.1 - Population Model Another population model is given...Ch. 2.1 - Terminal Velocity In Section 1.3 we saw that the...Ch. 2.1 - Suppose the model in Problem 40 is modified so...Ch. 2.1 - Chemical reactions When certain kinds of chemicals...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 122 solve the given differential...Ch. 2.2 - In Problems 2328 find an explicit solution of the...Ch. 2.2 - In Problems 2328 find an explicit solution of the...Ch. 2.2 - In Problems 2328 find an explicit solution of the...Ch. 2.2 - In Problems 2328 find an explicit solution of the...Ch. 2.2 - In Problems 2328 find an explicit solution of the...Ch. 2.2 - In Problems 2328 find an explicit solution of the...Ch. 2.2 - In Problems 29 and 30 proceed as in Example 5 and...Ch. 2.2 - In Problems 29 and 30 proceed as in Example 5 and...Ch. 2.2 - In Problems 3134 find an explicit solution of the...Ch. 2.2 - In Problems 3134 find an explicit solution of the...Ch. 2.2 - In Problems 3134 find an explicit solution of the...Ch. 2.2 - In Problems 3134 find an explicit solution of the...Ch. 2.2 - (a) Find a solution of the initial-value problem...Ch. 2.2 - Find a solution of xdydx=y2y that passes through...Ch. 2.2 - Find a singular solution of Problem 21. Of Problem...Ch. 2.2 - Show that an implicit solution of...Ch. 2.2 - Often a radical change in the form of the solution...Ch. 2.2 - Often a radical change in the form of the solution...Ch. 2.2 - Often a radical change in the form of the solution...Ch. 2.2 - Often a radical change in the form of the solution...Ch. 2.2 - Every autonomous first-order equation dy/dx = f(y)...Ch. 2.2 - (a) The autonomous first-order differential...Ch. 2.2 - In Problems 4550 use a technique of integration or...Ch. 2.2 - In Problems 4550 use a technique of integration or...Ch. 2.2 - In Problems 4550 use a technique of integration or...Ch. 2.2 - In Problems 4550 use a technique of integration or...Ch. 2.2 - In Problems 4550 use a technique of integration or...Ch. 2.2 - In Problems 4550 use a technique of integration or...Ch. 2.2 - Prob. 51ECh. 2.2 - Prob. 52ECh. 2.2 - In Problems 43 and 44 we saw that every autonomous...Ch. 2.2 - Prob. 54ECh. 2.2 - Find a function whose square plus the square of...Ch. 2.2 - Prob. 56ECh. 2.2 - Prob. 57ECh. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 1-24 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - Prob. 22ECh. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 124 find the general solution of the...Ch. 2.3 - In Problems 2536 solve the given initial-value...Ch. 2.3 - In Problems 2536 solve the given initial-value...Ch. 2.3 - In Problems 2536 solve the given initial-value...Ch. 2.3 - In Problems 2536 solve the given initial-value...Ch. 2.3 - In Problems 2536 solve the given initial-value...Ch. 2.3 - In Problems 2536 solve the given initial-value...Ch. 2.3 - In Problems 2536 solve the given initial-value...Ch. 2.3 - In Problems 2536 solve the given initial-value...Ch. 2.3 - In Problems 2536 solve the given initial-value...Ch. 2.3 - In Problems 2536 solve the given initial-value...Ch. 2.3 - In Problems 2536 solve the given initial-value...Ch. 2.3 - In Problems 2536 solve the given initial-value...Ch. 2.3 - In Problems 3740 proceed as in Example 6 to solve...Ch. 2.3 - In Problems 3740 proceed as in Example 6 to solve...Ch. 2.3 - In Problems 3740 proceed as in Example 6 to solve...Ch. 2.3 - In Problems 3740 proceed as in Example 6 to solve...Ch. 2.3 - In Problems 41 and 42 proceed as in Example 6 to...Ch. 2.3 - In Problems 41 and 42 proceed as in Example 6 to...Ch. 2.3 - In Problems 43 and 44 proceed as in Example 7 and...Ch. 2.3 - Prob. 44ECh. 2.3 - In Problems 45 and 46 proceed as in Example 7 and...Ch. 2.3 - Prob. 46ECh. 2.3 - Prob. 47ECh. 2.3 - Prob. 48ECh. 2.3 - Prob. 49ECh. 2.3 - Prob. 50ECh. 2.3 - Reread Example 4 and then find the general...Ch. 2.3 - Reread the discussion following Example 5....Ch. 2.3 - Prob. 53ECh. 2.3 - Prob. 54ECh. 2.3 - In determining the integrating factor (3), we did...Ch. 2.3 - Prob. 56ECh. 2.3 - Radioactive Decay Series The following system of...Ch. 2.3 - Heart Pacemaker A heart pacemaker consists of a...Ch. 2.3 - Prob. 61ECh. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In Problems 120 determine whether the given...Ch. 2.4 - In problems 2126 slove the given initial-value...Ch. 2.4 - In Problem 2126 solve the given initial-value...Ch. 2.4 - In problems 2126 solve the given intial-value...Ch. 2.4 - In Problems 2126 solve the given initial-value...Ch. 2.4 - In Problems 2126 solve the given initial-value...Ch. 2.4 - In Problems 2126 solve the given initial-value...Ch. 2.4 - In Problems 27 and 28 find the value of k so that...Ch. 2.4 - In Problems 27 and 28 find the value of k so that...Ch. 2.4 - In Problems 29 and 30 verify that the given...Ch. 2.4 - In Problems 29 and 30 verify that the given...Ch. 2.4 - In Problems 3136 solve the given differential...Ch. 2.4 - In Problems 3136 solve the given differential...Ch. 2.4 - In Problems 3136 solve the given differential...Ch. 2.4 - In Problems 3136 solve the given differential...Ch. 2.4 - In Problems 3136 solve the given differential...Ch. 2.4 - In Problems 3136 solve the given differential...Ch. 2.4 - In Problems 37 and 38 solve the given...Ch. 2.4 - In Problems 37 and 38 solve the given...Ch. 2.4 - (a) Show that a one-parameter family of solutions...Ch. 2.4 - Prob. 40ECh. 2.4 - Prob. 41ECh. 2.4 - Discuss how the functions M(x, y) and N(x, y) can...Ch. 2.4 - Differential equations are sometimes solved by...Ch. 2.4 - True or False: Every separable first-order...Ch. 2.4 - Falling Chain A portion of a uniform chain of...Ch. 2.5 - Each DE in Problems 114 is homogeneous. In...Ch. 2.5 - In Problems 110 solve the given differential...Ch. 2.5 - In Problems 110 solve the given differential...Ch. 2.5 - In Problems 1-10 solve the given differential...Ch. 2.5 - In Problems 110 solve the given differential...Ch. 2.5 - In Problems 1-10 solve the given differential...Ch. 2.5 - In Problems 110 solve the given differential...Ch. 2.5 - In Problems 110 solve the given differential...Ch. 2.5 - In Problems 110 solve the given differential...Ch. 2.5 - In Problems 110 solve the given differential...Ch. 2.5 - In Problems 1114 solve the given initial-value...Ch. 2.5 - In Problems 1114 solve the given initial-value...Ch. 2.5 - In Problems 1114 solve the given initial-value...Ch. 2.5 - In Problems 1114 solve the given initial-value...Ch. 2.5 - In Problems 1520 solve the given differential...Ch. 2.5 - In Problems 1520 solve the given differential...Ch. 2.5 - In Problems 1520 solve the given differential...Ch. 2.5 - In Problems 1520 solve the given differential...Ch. 2.5 - In Problems 1520 solve the given differential...Ch. 2.5 - In Problems 1520 solve the given differential...Ch. 2.5 - In Problems 21 and 22 solve the given...Ch. 2.5 - In Problems 21 and 22 solve the given...Ch. 2.5 - In Problems 2328 solve the given differential...Ch. 2.5 - In Problems 2328 solve the given differential...Ch. 2.5 - In Problems 2328 solve the given differential...Ch. 2.5 - In Problems 2328 solve the given differential...Ch. 2.5 - In Problems 2328 solve the given differential...Ch. 2.5 - In Problems 2328 solve the given differential...Ch. 2.5 - dydx=cos(x+y), y(0) = /4Ch. 2.5 - In Problems 29 and 30 solve the given...Ch. 2.5 - Explain why it is always possible to express any...Ch. 2.5 - Put the homogeneous differential equation...Ch. 2.5 - Prob. 33ECh. 2.5 - Prob. 34ECh. 2.5 - The differential equation dy/dx = P(x) + Q(x)y +...Ch. 2.5 - Determine an appropriate substitution to solve...Ch. 2.5 - Falling Chain In Problem 45 in Exercises 2.4 we...Ch. 2.5 - Population Growth In the study of population...Ch. 2.6 - In Problems 1 and 2 use Eulers method to obtain a...Ch. 2.6 - In Problems 1 and 2 use Eulers method to obtain a...Ch. 2.6 - In Problems 3 and 4 use Eulers method to obtain a...Ch. 2.6 - In Problems 3 and 4 use Eulers method to obtain a...Ch. 2.6 - In Problems 510 use a numerical solver and Eulers...Ch. 2.6 - In Problems 510 use a numerical solver and Eulers...Ch. 2.6 - In Problems 510 use a numerical solver and Eulers...Ch. 2.6 - In Problems 510 use a numerical solver and Eulers...Ch. 2.6 - In Problems 510 use a numerical solver and Eulers...Ch. 2.6 - In Problems 510 use a numerical solver and Eulers...Ch. 2 - Answer Problems 112 without referring back to the...Ch. 2 - Prob. 2RECh. 2 - Prob. 3RECh. 2 - Prob. 4RECh. 2 - Prob. 5RECh. 2 - Prob. 6RECh. 2 - Prob. 7RECh. 2 - Prob. 8RECh. 2 - Prob. 9RECh. 2 - Prob. 10RECh. 2 - Prob. 11RECh. 2 - Prob. 12RECh. 2 - Prob. 13RECh. 2 - In Problems 13 and 14 construct an autonomous...Ch. 2 - The number 0 is a critical point of the autonomous...Ch. 2 - Prob. 16RECh. 2 - Prob. 17RECh. 2 - Classify each differential equation as separable,...Ch. 2 - In Problems 1926 solve the given differential...Ch. 2 - In Problems 1926 solve the given differential...Ch. 2 - Prob. 21RECh. 2 - In Problems 1926 solve the given differential...Ch. 2 - In Problems 1926 solve the given differential...Ch. 2 - In Problems 1926 solve the given differential...Ch. 2 - In Problems 1926 solve the given differential...Ch. 2 - In Problems 1926 solve the given differential...Ch. 2 - In Problems 2730 express the solution of the given...Ch. 2 - In Problems 2730 express the solution of the given...Ch. 2 - Prob. 29RECh. 2 - Prob. 30RECh. 2 - Prob. 31RECh. 2 - In Problems 31 and 32 solve the given...Ch. 2 - Prob. 33RECh. 2 - Prob. 34RECh. 2 - (a) Without solving, explain why the initial-value...Ch. 2 - (a) Find an implicit solution of the initial-value...Ch. 2 - Prob. 37RECh. 2 - Use Eulers method with step size h = 0.1 to...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.Similar questions
- Find all positive integers n such that n.2n +1 is a square.arrow_forwardA researcher wishes to estimate, with 90% confidence, the population proportion of adults who support labeling legislation for genetically modified organisms (GMOs). Her estimate must be accurate within 4% of the true proportion. (a) No preliminary estimate is available. Find the minimum sample size needed. (b) Find the minimum sample size needed, using a prior study that found that 65% of the respondents said they support labeling legislation for GMOs. (c) Compare the results from parts (a) and (b). ... (a) What is the minimum sample size needed assuming that no prior information is available? n = (Round up to the nearest whole number as needed.)arrow_forwardThe table available below shows the costs per mile (in cents) for a sample of automobiles. At a = 0.05, can you conclude that at least one mean cost per mile is different from the others? Click on the icon to view the data table. Let Hss, HMS, HLS, Hsuv and Hмy represent the mean costs per mile for small sedans, medium sedans, large sedans, SUV 4WDs, and minivans respectively. What are the hypotheses for this test? OA. Ho: Not all the means are equal. Ha Hss HMS HLS HSUV HMV B. Ho Hss HMS HLS HSUV = μMV Ha: Hss *HMS *HLS*HSUV * HMV C. Ho Hss HMS HLS HSUV =μMV = = H: Not all the means are equal. D. Ho Hss HMS HLS HSUV HMV Ha Hss HMS HLS =HSUV = HMVarrow_forward
- Question: A company launches two different marketing campaigns to promote the same product in two different regions. After one month, the company collects the sales data (in units sold) from both regions to compare the effectiveness of the campaigns. The company wants to determine whether there is a significant difference in the mean sales between the two regions. Perform a two sample T-test You can provide your answer by inserting a text box and the answer must include: Null hypothesis, Alternative hypothesis, Show answer (output table/summary table), and Conclusion based on the P value. (2 points = 0.5 x 4 Answers) Each of these is worth 0.5 points. However, showing the calculation is must. If calculation is missing, the whole answer won't get any credit.arrow_forwardBinomial Prob. Question: A new teaching method claims to improve student engagement. A survey reveals that 60% of students find this method engaging. If 15 students are randomly selected, what is the probability that: a) Exactly 9 students find the method engaging?b) At least 7 students find the method engaging? (2 points = 1 x 2 answers) Provide answers in the yellow cellsarrow_forwardIn a survey of 2273 adults, 739 say they believe in UFOS. Construct a 95% confidence interval for the population proportion of adults who believe in UFOs. A 95% confidence interval for the population proportion is ( ☐, ☐ ). (Round to three decimal places as needed.)arrow_forward
- Find the minimum sample size n needed to estimate μ for the given values of c, σ, and E. C=0.98, σ 6.7, and E = 2 Assume that a preliminary sample has at least 30 members. n = (Round up to the nearest whole number.)arrow_forwardIn a survey of 2193 adults in a recent year, 1233 say they have made a New Year's resolution. Construct 90% and 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals. The 90% confidence interval for the population proportion p is (Round to three decimal places as needed.) J.D) .arrow_forwardLet p be the population proportion for the following condition. Find the point estimates for p and q. In a survey of 1143 adults from country A, 317 said that they were not confident that the food they eat in country A is safe. The point estimate for p, p, is (Round to three decimal places as needed.) ...arrow_forward
- A straight-line H is tangent to the function g(x)=-6x-3+ 8 and passes through the point (- 4,7). Determine, the gradient of the straight-line Choose.... y-intercept of the straight-line Choose... + which of the following is the answers -1.125 -6.72 1.125 7.28 0.07 - 7.28 6.72arrow_forwardCan you help me find the result of an integral + a 炉[メをメ +炉なarrow_forward2 a Can you help me find the result of an integral a 아 x² dxarrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Discrete Mathematics and Its Applications ( 8th I...MathISBN:9781259676512Author:Kenneth H RosenPublisher:McGraw-Hill EducationMathematics for Elementary Teachers with Activiti...MathISBN:9780134392790Author:Beckmann, SybillaPublisher:PEARSON
- Thinking Mathematically (7th Edition)MathISBN:9780134683713Author:Robert F. BlitzerPublisher:PEARSONDiscrete Mathematics With ApplicationsMathISBN:9781337694193Author:EPP, Susanna S.Publisher:Cengage Learning,Pathways To Math Literacy (looseleaf)MathISBN:9781259985607Author:David Sobecki Professor, Brian A. MercerPublisher:McGraw-Hill Education

Discrete Mathematics and Its Applications ( 8th I...
Math
ISBN:9781259676512
Author:Kenneth H Rosen
Publisher:McGraw-Hill Education

Mathematics for Elementary Teachers with Activiti...
Math
ISBN:9780134392790
Author:Beckmann, Sybilla
Publisher:PEARSON


Thinking Mathematically (7th Edition)
Math
ISBN:9780134683713
Author:Robert F. Blitzer
Publisher:PEARSON

Discrete Mathematics With Applications
Math
ISBN:9781337694193
Author:EPP, Susanna S.
Publisher:Cengage Learning,

Pathways To Math Literacy (looseleaf)
Math
ISBN:9781259985607
Author:David Sobecki Professor, Brian A. Mercer
Publisher:McGraw-Hill Education
01 - What Is A Differential Equation in Calculus? Learn to Solve Ordinary Differential Equations.; Author: Math and Science;https://www.youtube.com/watch?v=K80YEHQpx9g;License: Standard YouTube License, CC-BY
Higher Order Differential Equation with constant coefficient (GATE) (Part 1) l GATE 2018; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=ODxP7BbqAjA;License: Standard YouTube License, CC-BY
Solution of Differential Equations and Initial Value Problems; Author: Jefril Amboy;https://www.youtube.com/watch?v=Q68sk7XS-dc;License: Standard YouTube License, CC-BY