Question is in image provided Find the general solution of the following differential equation. Primes denote derivatives with respect to x.x*y' + 7xy = 15y3The general solution is(Type an implicit general solution in the form F(x,y) = C, where C is an arbitrary constant. Type an expression using x and y as the variables.)

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Description: Question is in image provided Find the general solution of the following differential equation. Primes denote derivatives with respect to x.x*y' + 7xy = 15y3The general solution is(Type an implicit general solution in the form F(x,y) = C, where C is

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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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Find the general solution of the following differential equation. Primes denote derivatives with respect to x.
x*y' + 7xy = 15y3
The general solution is
(Type an implicit general solution in the form F(x,y) = C, where C is an arbitrary constant. Type an expression using x and y as the variables.)

Expert Solution

Step 1

Given differential equation is $x^{2} y^{'} + 7 x y = 15 y^{3}$

It can be written as follows

$x^{2} \frac{d y}{d x} + 7 x y = 15 y^{3}$

Divide both sides by $x^{2} y^{3}$

$\begin{matrix} \frac{x^{2}}{x^{2} y^{3}} \frac{d y}{d x} + \frac{7 x y}{x^{2} y^{3}} = \frac{15 y^{3}}{x^{2} y^{3}} \\ \frac{1}{y^{3}} \frac{d y}{d x} + \frac{7}{x} \frac{1}{y^{2}} = \frac{15}{x^{2}} \end{matrix}$

Substitute $\frac{1}{y^{2}} = v$

$\begin{array}{l} \Rightarrow - 2 \frac{1}{y^{3}} d y = d v \\ \Rightarrow \frac{1}{y^{3}} \frac{d y}{d x} = - \frac{1}{2} \frac{d v}{d x} \end{array}$

Substitute the values

$\begin{matrix} - \frac{1}{2} \frac{d v}{d x} + \frac{7}{x} v = \frac{15}{x^{2}} \\ \frac{d v}{d x} - \frac{14}{x} v = - \frac{30}{x^{2}} (Multiply both sides by - 2) \end{matrix}$

Hence, the differential equation $\frac{d v}{d x} - \frac{14}{x} v = - \frac{30}{x^{2}}$ is linear in $v$ and of the form $\frac{d y}{d x} + P (x) y = Q (x)$

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