Question

find the solution of the differential equation (1+x2) dy/dx x=0
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Answer 1

Step-by-step explanation:

The given differential equation is

$\quad (1+x^{2})\frac{dy}{dx}+x=0$

$\Rightarrow \frac{dy}{dx}+\frac{x}{1+x^{2}}=0,\:since\:1+x^{2}\neq 0$

$\Rightarrow dy+\frac{x}{1+x^{2}}dx=0$

$\Rightarrow dy+\frac{1}{2}\:\frac{2x}{1+x^{2}}dx=0$

$\Rightarrow dy+\frac{1}{2}\:d\{log(1+x^{2})\}=0$

Taking integration on both sides, we get -

$\quad \int dy+\frac{1}{2}\int d\{log(1+x^{2})\}=c$ where c is constant of integration

$\Rightarrow y+\frac{1}{2}\:log(1+x^{2})=c$

This is the required integral.

Answer:

The solution of the differential equation

$\quad (1+x^{2})\frac{dy}{dx}+x=0$

is

$\quad y+\frac{1}{2}\:log(1+x^{2})=c$.