Painevg

Answered

2021-12-31

Solve for G.S. / P.S. for the following differential equations using the method of solution for homogeneous equations.

$-(3x+25y)dx+(25x+3y)dy=0$

Answer & Explanation

servidopolisxv

Expert

2022-01-01Added 27 answers

Step 1

Given differential equation is$-(3x+25y)dx+(25x+3y)dy=0$

Given differential equation can be written as:

$\frac{dy}{dx}=\frac{3x+25y}{25x+3y}$

This homogeneous differential equation.

Put$y=vx$

$\frac{dy}{dx}=v+x\frac{dv}{dx}$

Step 2

Substituting the value, we get

$v+x\frac{dv}{dx}=\frac{3x+25vx}{25x+3vx}$

$v+x\frac{dv}{dx}=\frac{3+25v}{25+3v}$

$x\frac{dv}{dx}=\frac{3+25v}{25+3v}-v$

$x\frac{dv}{dx}=\frac{3+25v-v(25+3v)}{25+3v}$

$x\frac{dv}{dx}=\frac{3-3{v}^{2}}{25+3v}$

$\left(\frac{25+3v}{3-3{v}^{2}}\right)dv=\frac{1}{x}dx$

Step 3

Integrating both sides, we get

$\int \left(\frac{25+3v}{3-3{v}^{2}}\right)dv=\int \frac{1}{x}dx$

$\frac{1}{3}[\int \frac{25}{(1-{v}^{2})}dv+\int \frac{3v}{(1-{v}^{2})}dv]=\mathrm{ln}x+c$

$\frac{1}{3}[\frac{25}{2}\mathrm{ln}\left(\frac{1+v}{1-v}\right)-\frac{3}{2}\mathrm{ln}(1-{v}^{2})]=\mathrm{ln}x+c$

Given differential equation is

Given differential equation can be written as:

This homogeneous differential equation.

Put

Step 2

Substituting the value, we get

Step 3

Integrating both sides, we get

Jim Hunt

Expert

2022-01-02Added 45 answers

Let

Take, integration both sides

Take,

Take,

Vasquez

Expert

2022-01-09Added 457 answers

Most Popular Questions