(1 point) Solve the following differential equation:dy.-dr +1- a?y?dx =1- r²y?= constant. help (formulas)

Given that dx=y1-x2y2dx+x1-x2y2dy The objective is to find the solution for the differential…

Answered: -dx + 1 - x²y? dy. 1– a²y? dx =…

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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(1 point) Solve the following differential equation:
dy.
-dr +
1- a?y?
dx =
1- r²y?
= constant. help (formulas)

Expert Solution

Step 1

Given that $d x = \frac{y}{1 - x^{2} y^{2}} d x + \frac{x}{1 - x^{2} y^{2}} d y$

The objective is to find the solution for the differential equation.

The differential equation can be rewritten as,

$(1 - x^{2} y^{2}) d x = y d x + x d y (1 - x^{2} y^{2}) d x - d (x y) = 0$

The equation is of the form $M d x + N d (x y) = 0$ .

So let's find $M_{x y}$ for the given differential equation,

$\begin{array}{rcl} M_{x y} & = & 0 - 2 x y \\ = & - 2 x y \end{array}$

and find Nx for the given differential equation,

$N_{x} = 0$

Since $M_{x y} \neq N_{x}$ the equation is not exact.

So let's find the integrating factor,

The formula to find the integrating factor is,

$\begin{array}{rcl} I F & = & e^{\int \frac{N_{x} - M_{x y}}{M} d x} \\ = & e^{\int \frac{2 x y}{1 - x^{2} y^{2}} d x y} \\ = & e^{\ln (\frac{1}{1 - x^{2} y^{2}})} \\ = & \frac{1}{1 - x^{2} y^{2}} \end{array}$

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