(7) (a) Solve the initial value problem z²y + 2xy = 6x³, y(1) = 1.(b) Use direction field to verify the solution you got is correct. Show the direction field plot.

Answered: (7) (a) Solve the initial value problem…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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(7) (a) Solve the initial value problem z²y + 2xy = 6x³, y(1) = 1.
(b) Use direction field to verify the solution you got is correct. Show the direction field plot.

Expert Solution

Step 1

What is Initial Value Problem:

An ordinary differential equation follows a general solution that contains one or more than integrating constant. As it is an application of integral calculus, integration results the integrating constant. But, if we wish to determine a particular solution that belongs to the family of solution curves and passes through the initial points, we can determine the constant and it gives the particular solution.

Given:

Given initial value problem is

$x^{2} y' + 2 x y = 6 x^{5}$

With initial condition $y (1) = 1$ .

To Determine:

We solve the problem and draw the slope field.