Answer to Question #122623 in Differential Equations for NILAM JYOTI DAS

Question

Write the Integrating factor of the Differential Equation dy/dx
+ x/1+x *
y=ex

Answer 1

$\frac {dy}{dx} + \frac {x}{1+x} y = e^x$

Comparing with $\frac {dy}{dx} + Py = Q$ we get

P = $\frac {x}{1+x}$ = $\frac {x}{x+1}$ and Q = e^x

So integrating factor is

$e^{\int Pdx}$ = $e^{\int \frac {x}{x+1}dx}$ = $e^{\int \frac {x+1-1}{x+1}dx}$

= $e^{\int ( \frac {x+1}{x+1} - \frac {1}{x+1})dx}$

= $e^{\int ( 1 - \frac {1}{x+1})dx}$

= $e^{x - ln(1+x)}$

= $\frac {e^x}{e^{ln(1+x)}}$

= $\frac {e^x}{1+x}$

$\mathbf {Answer}$

The integrating factor is $\frac {e^x}{1+x}$