Question

Solve the differential equation by substitution suggested by equation. Show complete solution.

du/dv =(u-v)^2 - 2(u-v) - 2

Answer 1

Solution;

$\frac{du}{dv}=(u-v)^2-2(u-v)-2$

Take ;

$x=(u-v)$

Such that;

$\frac{dx}{dv}=\frac{du}{dv}-1$

Substitute into the equation;

$\frac{dx}{dv}+1=x^2-2x-2$

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

Seperate by variables;

$\frac{1}{x^2-2x-3}dx=dv$

Integrating;

$\frac{ln(|x-3|)-ln(|x+1|)}{4}=v+c$

But ;

x=u-v;

Replace back;

$\frac{ln(|u-v-3|)-ln(|u-v+1|)}{4}=v+C$