### Differential Equations and Particular SolutionsIn this section, we explore different forms of particular solutions for various differential equations, labeled from A to F. Each form is used based on the non-homogeneous part of the differential equation.#### Forms of Particular Solutions:- **A.** $ y_p(x) = Ax^2 + Bx + C $- **B.** $ y_p(x) = Ae^{2x} $- **C.** $ y_p(x) = A \cos 2x + B \sin 2x $- **D.** $ y_p(x) = (Ax + B) \cos 2x + (Cx + D) \sin 2x $- **E.** $ y_p(x) = Axe^{2x} $- **F.** $ y_p(x) = e^{3x} (A \cos 2x + B \sin 2x) $#### Differential Equations:Determine the form of the particular solution for each given differential equation:1. $ \frac{d^2 y}{dx^2} + 4y = x - \frac{x^2}{20} $2. $ \frac{d^2 y}{dx^2} + 6 \frac{dy}{dx} + 8y = e^{2x} $3. $ y'' + 4y' + 13y = 3 \cos 2x $4. $ y'' - 2y' - 15y = e^{3x} \cos 2x $ ### Explanation:For each differential equation, the particular solution is chosen based on the form of the non-homogeneous part. For instance:- **Equation 1** involves polynomial terms, so a polynomial form like **A** may be suitable.- **Equation 2** features an exponential term $ e^{2x} $, making form **B** appropriate.- **Equation 3** involves trigonometric functions $ \cos 2x $, suggesting form **C** or **D** depending on whether additional terms are needed.- **Equation 4** combines exponential and trigonometric terms $ e^{3x} \cos 2x $, fitting form **F**. These forms assist in constructing a particular solution for each differential equation, providing

Answered: Image /qna-images/answer/1c0fc3a6-441b-44ca-866b-f2d8c08b4eaf.jpg

A. Yp(x) = Ax² + Bx + C, B. Y,(x) = Ae², c. y,(x)= Acos 2x + Bsin 2z D. yp(x) = (Ax +B) cos 2x + (Cx + D) sin 2x E. Y½(x) = Axe", and F. Yp(x) = e³" (A cos 2x + B sin 2z) %3D d²y + 4y = x dz? 20 d²y dy +6 + 8y = e²z dr dr? y" + 4y + 13y = 3 cos 2z y" – 2y' – 15y = e cos 2x

A. Yp(x) = Ax² + Bx + C, B. Y,(x) = Ae², c. y,(x)= Acos 2x + Bsin 2z D. yp(x) = (Ax +B) cos 2x + (Cx + D) sin 2x E. Y½(x) = Axe", and F. Yp(x) = e³" (A cos 2x + B sin 2z) %3D d²y + 4y = x dz? 20 d²y dy +6 + 8y = e²z dr dr? y" + 4y + 13y = 3 cos 2z y" – 2y' – 15y = e cos 2x

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: f= x? + क) | + ला (i) of =? (üi, af मु] liv) of

Q: Suppose f(x) = sin 4-x). Evaluate: x) dx (A - 3cos FC B -cos 3 X-4+ C 3 Cos 3 (D) 3cos -x- 4)+C

Q: Let ro)-플-x a-글 FW) = b = 0, and n=3. Then: cos (f(x))dx =

A: We can find the integration as below

Q: 2) If y = cos x, what is y (®(x)?

Q: The general solution of y" + y = e + 5e2* is y = ei (A cos (*) + B sin )) + Ce-" + 응e2x 2 () V5z y =…

A: Given that y"'+y=e^(-x)+5e^(2x) To find the general solution

Q: Q3) Simplify (if necessary) then Compute: y"+ 2y Where y(x) = (zx+ cosx*)] [4(cosh2x-sinh2x)] d 2,2…

A: To find - Simplify (if necessary) then Compute : y'' + 2y' where yx = ∂2∂x ∂zz2x + cosx44cosh2x -…

Q: Find the form for a particular solution to the equation y "+2y'– 3y =r?cos (nt). A…

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: +3x2 B) Find dy/dx if : (****")) (v)² = sin (cos³

A: Differentiation chain rule

Q: dy 5- dx for y = tan (3x) is equal to: b) sec (3x) a) 6x sec(3x) tan(3x) c) 6x sec (3x) 6- dx for y…

Q: Find dy Y= #an" (=) + cos "(sx) + CoS

Q: Suppose that 3x² - y² = t² and y=tsin 0. Find (27). = sin e 0 X = 0 and

A: Given 3x2-y2=t2 and y=tsinθ

Q: 8 410x-32+x2 cos2x 4 Find *: (a) y =

Q: The function f(x) = a sin x + b cos x + c satisfies f"(x) = cos x – 2 sin x, f'(0) = f(0) = 2. Find…

A: a=2 b=-1 c=3

Q: If x²y" + xy' + (x² – ÷)y=o x >0 ,V,=x 2 sinx , and y =y;v. Which of the following equations…

Q: 8. Differentiate with respect to x: (a) y = sin(x³), (b) y = sin³ x and (c) y = (7- cos(5x)) 2023

Q: 3. Find the derivatives: a) f(x) = 3x³-4 2x²+3 b) DO NOT SIMPLIFY y = (x² + 6x + 2)(²+√x + 3) c)…

A: Given; a) f(x)=(3x3-4)/(2x2+3) b) y= (x2+6x+2)(2/x4 + 3sqrt(x5) +3) c) g(x)=-2csc7x+8sin3x+12tan2x…

Q: Q# (13 Rewrite: cos (x)sin(x)dx A) ſ u'du 1/2 B) Tu°du V312 C) [ -u°du 1 V3/2 D) [ u°du 1 V3/2 E)…

Q: A power signal which is periodic with period 2n and is characterized by the form f(x) Expand f(x) in…

A: Given, fx=sinhxcosx, -π≤x≤π Since, f is periodic of period 2π then, fx+2π=fx Then,…

Q: Q1) Evaluate: (a) fe²_1_dx Inx sin ¹x 1-x² dx (b) f 1- cos dx (e) Jỗ 2sinx cosx dx (c) | Inx dx

Q: Given z(x, y) =In(2xy) where x(u,v)= e" cos(4v) and y(u,v)=e" sin(4v). Find ди 1 cos (4r)+-sin(4v)…

Q: If g(x) = 3 sin x + 2 cos x + 5, then g' (3) = A B 12/3-√3 - 3/2+√3

Q: Q3) A: Find y if: a- y = sin 2x + 2 cos x c- y = sin √x (As... b- y = secx. csc x d- y = tan(2x + 1)

A: Please see the attachment

Q: d (c) (cos(2²) + In(3 – 2æ)) dx

Q: dy Find if y = 4 sin?(3x) dx

A: Here we have to find the derivative of function y which is given as above. Here I am attaching a…

Q: // (cos (a? + y³)dA where D: {(x, y) | 4 < x² + y² < 49} Find

Q: Let g(x) - S²³ = 2 sin(t)e dt. Then g'(x) equals

A: Here for given g(x) we have to find g'(x)

Q: (D4 + 18D2 + 81)y = 0 (A y = (c, + c,) cos3x + (c,+c,) sin3x. (B у %3 (с, + сх) cos:3x + (с,+c)…

A: To solve the given differential equation first we construct auxiliary equation and then solving the…

Q: 1. (D² - 4D+5) (e² sin x)

Q: In implicit differentation problems it seems we treat x and y differently. For example, when we…

A: Recall: Chain Rule of differentiation : If F(x)=f(g(x)) then , F'(x) = df(g(x))dx =…

Q: COs ax dx. x² + b² (q)

A: Cauchy residue theorem: If f is holomorphic function defined on simply connected open set with…

Q: 4. Consider a3(x)y"" + a2(x)y" + a₁(x)y' = 0 (a) What's one trivial solution? (b) Suppose the…

A: Please see the attachment

Q: 5 Find *: (a) y = cos(11x + 2x) dy

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

### Differential Equations and Particular Solutions

In this section, we explore different forms of particular solutions for various differential equations, labeled from A to F. Each form is used based on the non-homogeneous part of the differential equation.

#### Forms of Particular Solutions:
- **A.** $ y_p(x) = Ax^2 + Bx + C $
- **B.** $ y_p(x) = Ae^{2x} $
- **C.** $ y_p(x) = A \cos 2x + B \sin 2x $
- **D.** $ y_p(x) = (Ax + B) \cos 2x + (Cx + D) \sin 2x $
- **E.** $ y_p(x) = Axe^{2x} $
- **F.** $ y_p(x) = e^{3x} (A \cos 2x + B \sin 2x) $

#### Differential Equations:
Determine the form of the particular solution for each given differential equation:

1. $ \frac{d^2 y}{dx^2} + 4y = x - \frac{x^2}{20} $
2. $ \frac{d^2 y}{dx^2} + 6 \frac{dy}{dx} + 8y = e^{2x} $
3. $ y'' + 4y' + 13y = 3 \cos 2x $
4. $ y'' - 2y' - 15y = e^{3x} \cos 2x $

### Explanation:

For each differential equation, the particular solution is chosen based on the form of the non-homogeneous part. For instance:

- **Equation 1** involves polynomial terms, so a polynomial form like **A** may be suitable.
- **Equation 2** features an exponential term $ e^{2x} $, making form **B** appropriate.
- **Equation 3** involves trigonometric functions $ \cos 2x $, suggesting form **C** or **D** depending on whether additional terms are needed.
- **Equation 4** combines exponential and trigonometric terms $ e^{3x} \cos 2x $, fitting form **F**.

These forms assist in constructing a particular solution for each differential equation, providing

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Trending now

This is a popular solution!

Step by step

Solved in 5 steps with 5 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Differential Equation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

2) x³y? = 2x + 2y3 3) y = sin?(x) - cos³(2x) + 2
IM Oy O (2n) (a)ea (b) cos 3 (d) sin 3 (e) None (c) cos a
(b) Find (f ¹)'(1) if f(x) = x² +5 sin " + cos2. (TX) (X)
x3 a) 8 x4 dx | b) (tan(5æ) + sec(5x)) dr c) 8sin(x) cos(x) dx
2.) (3x + x2) sin(2x) dx Use whatever method you want to
// (cos (x² + y³)dA where D = {(x, y) | 16 < 2² + y? < 64} Find
I need the answer as soon as possible
36) y = 2 sin x - 4 wans To tremsinte 104y 8+ +- TC 3T -8 -10 2.