Let y=y(x) be the solution of the differential equation (3y^2-5x^2)ydx+2x(x^2-y^2)dy=0 suc…

Question

Let $y=y(x)$ be the solution of the differential equation

$(3{y}^{2}-5{x}^{2})ydx+2x({x}^{2}-{y}^{2})dy=0$

such that $y(1)=1$. Then $|(y(2){)}^{3}-12y(2)|$ is equal to :

Accepted Answer

$(3{y}^{2}-5{x}^{2})y\cdot dx+2x({x}^{2}-{y}^{2})dy=0$

$\Rightarrow \frac{dy}{dx}=\frac{y(5{x}^{2}-3{y}^{2})}{2x({x}^{2}-{y}^{2})}$

Put $y=mx$

$\Rightarrow m+x\cdot \frac{dm}{dx}=\frac{m(5-3{m}^{2})}{2(1-{m}^{2})}$

$\begin{gathered} x\cdot \frac{dm}{dx}=\frac{(5-3{m}^{2})m-2m(1-{m}^{2})}{2(1-{m}^{2})} \ \Rightarrow \frac{dx}{x}=\frac{2({m}^{2}-1)}{m({m}^{2}-3)}dm \ \Rightarrow \frac{dx}{x}=(\frac{2}{m}-\frac{\frac{4}{3}}{m}+\frac{\frac{4m}{3}}{{m}^{2}-3})dm \end{gathered}$

$\Rightarrow \int \frac{dx}{x}=\int \frac{(\frac{2}{3})}{m}+\int \frac{2}{3}(\frac{2m}{{m}^{2}-3})dm$

$\Rightarrow \ln |x|=\frac{2}{3}\ln |m|+\frac{2}{3}\ln |{m}^{2}-3|+C$

Or, $\ln |x|=\frac{2}{3}\ln |\frac{y}{x}|+\frac{2}{3}\ln |{(\frac{y}{x})}^{2}-3|+C$

Put $(x=1,y=1)$ : we get $c=-\frac{2}{3}\ln (2)$

$\Rightarrow \ln |x|=\frac{2}{3}\ln |\frac{y}{x}|+\frac{2}{3}\ln |{(\frac{y}{x})}^{2}-3|-\frac{2}{3}\ln (2)$

$\Rightarrow (\frac{y}{x})[{(\frac{y}{x})}^{2}-3]=2.({x}^{3/2})$ ........(1)

Put $x=2$ in equation (1), we get

$\Rightarrow y({y}^{2}-12)=4	imes 2	imes 2	imes 2\sqrt{2}$

$\Rightarrow {y}^{3}-12y=32\sqrt{2}$

$\Rightarrow |{y}^{3}(2)-12y(2)|=32\sqrt{2}$

Solutions · 1