The solution of the differential equation [dy dx = x + y x] satisfying the condition y(1)=…

Question

The solution of the differential equation

$${{dy} \over {dx}} = {{x + y} \over x}$$ satisfying the condition $y(1)=1$ is :

Accepted Answer

$${{dy} \over {dx}} = {{x + y} \over x} = 1 + {y \over x}$$

Putting $y=$ $vx$

and $${{dy} \over {dx}} = v + x{{dv} \over {dx}}$$

we get

$$v + x{{dv} \over {dx}} = 1 + v$$

$$ \Rightarrow \int {{{dx} \over x}}  = \int {dv} $$

$$ \Rightarrow v = \ln {\mkern 1mu} x + c$$

$$ \Rightarrow y = x\ln x + cx$$

As $$\,\,\,\,y\left( 1 \right) = 1$$

$$\therefore$$ $c=1$

So solution is $y=xlnx+x$

Solutions · 1