13 -6.7.als als alsaladxdydxdydxdyey + eyx²dx1+ y²X-X8.yIn each of Problems 9 through 16:a. Find the solution of the given initial value problem in explicitform.Gb. Plot the graph of the solution.c. Determine (at least approximately) the interval in which thesolution is defined.9. y'=(1-2x) y², y(0) = -1/610.y'=(1-2x)/y, y(1) = -211.xdx+ye dy = 0, y(0) = 112. dr/d0 = r2²/0, r(1) = 213.14. y'= 2x/(1+2y), y(2) = 0lo15. y'= (3x² - e*)/(2y-5), y(0) = 116. sin(2x) dx + cos(3y) dy = 0, y(π/2) = π/3Some of the results requested in Problems 17 through 22 can beobtained either by solving the given equations analytically or byplotting numerically generated approximations to the solutions. Tryto form an opinion about the advantages and disadvantages of eachapproach.G 17. Solve the initial value problemy'=xy³ (1+x²)-1/2, y(0) = 1y' =1+3x²Зу2 — буy' =y(0) = 1and determine the interval in which the solution is valid.Hint: To find the interval of definition, look for points where theintegral curve has a vertical tangent.G 18. Solve the initial value problem3.x²3y² - 4' y(1) = 0and determine the interval in which the solution is valid.Hint: To find the interval of definition, look for points where theintegral curve has a vertical tangent.G 2121.y'a. Determine howdepends on the initiab. Suppose that yofirst reaches the valuG 22. Consider the intyy'a.Determine howb. If yo = 2, find tthe value 3.99.c. Find the range cthe interval 3.99<23. Solve the equationwhere a, b, c, and d are24. Use separation ofdQdtwhere a, b, r, and Qobehaves as t-xHomogeneous Equaticdy/dx = f(x, y) canonly, then the equationcan always be transformthe dependent variable. Ihomogeneous equationsThe word "homogeneouscontexts. The homogeneouthe homogeneous equation

Answered: 13. y'=xy³ (1+x²)-1/2, y(0) = 1

7th Edition

ISBN:9780133594140

Author:James Kurose, Keith Ross

Publisher:James Kurose, Keith Ross

Chapter1: Computer Networks And The Internet

Section: Chapter Questions

Problem R1RQ: What is the difference between a host and an end system? List several different types of end...

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Expert Solution

Step 1

Given:-

y' = xy^3(1+x^2)^(-1/2) , y(0) =1

ans :-

Here's a solution to the differential equation y' = xy^3(1+x^2)^(-1/2) using separation of variables:-

Separate the variables:

dy/dx = xy^3(1 + x^2)^(-1/2)
(1 + x^2)^(1/2) dy = xy^3 dx
Integrate both sides:

∫(1 + x^2)^(1/2) dy = ∫xy^3 dx
y * (1 + x^2)^(1/2) = (2/4) x^4 y^4 + C
Solve for y:

y^4 * (1 + x^2) = Cx^2 + C'
y^4 = (Cx^2 + C') / (1 + x^2)