Question

For the differential equation $(1+y+x^{2}y)dx+(x+x^3)dy=0.$ If $y\left(\frac{\pi }{4}\right)=0$ then find the absolute value of the constant.

Answer 1

$(1+y+x^{2}y)dx+(x+x^3)dy=0$
$\frac{dy}{dx}=\frac{-1}{x(1+x^2)}-\frac{y}{x}$
$\Rightarrow \frac{dy}{dx}+\frac{y}{x}=\frac{-1}{x(1+x^2)}$
which is a linear differential eqn with y as dependent variable.
Here, $P=\frac{1}{x};Q=-\frac{1}{x(1+x^2)}$
Integrating factor $I.F.=e^{Pdx}$
$=e^{\int \left(\frac{1}{x}\right)dx}$
$=e^{\log x}$
$\Rightarrow I.F.=x$
Solution of given differential eqn is
$yx=\int x\frac{-1}{x(1+x^2)dx}$
$\Rightarrow xy={\cot }^{-1}x+C$
Given $y(\frac{\pi }{4})=0$
$\Rightarrow 0={\cot }^{-1}\frac{\pi }{4}+C$
$\Rightarrow C=-{\cot }^{-1}\frac{\pi }{4}$
$|C|=1$