2. Consider the the initial value problem.xe*+v° dx + y(e*+ + 1) dy = 0, y(0) = 0%3|(a) Is this differential equation is exact ? If no, find the integrating factor.(b) Solve the initial value problem.

Answered: 2. Consider the the initial value…

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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2. Consider the the initial value problem.
xe*+v° dx + y(e*+ + 1) dy = 0, y(0) = 0
%3|
(a) Is this differential equation is exact ? If no, find the integrating factor.
(b) Solve the initial value problem.

Expert Solution

Step 1

the given equation is $x e^{x^{2} + y^{2}} d x + y (e^{x^{2} + y^{2}} + 1) d y = 0$

The given equation is of the form $M d x + N d y = 0$

The equation is exact if $M_{y} = N_{x}$

Here $M = x (e^{x^{2} + y^{2}})$

$N = y (e^{x^{2} + y^{2}} + 1)$

find $M_{y}$

$\begin{array}{rcl} M_{y} & = & \frac{\partial}{\partial y} x (e^{x^{2} + y^{2}}) \\ = & x e^{x^{2} + y^{2}} 2 y \\ = & 2 x y e^{x^{2} + y^{2}} \end{array}$

Now find $\begin{array}{rcl} N_{x} \end{array}$

$\begin{array}{rcl} N_{x} & = & \frac{\partial}{\partial x} y (e^{x^{2} + y^{2}} + 1) \\ = & y (e^{x^{2} + y^{2}}) 2 x \\ = & 2 x y (e^{x^{2} + y^{2}}) \end{array}$

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