Question

The solution of the differential equation $\frac{dy}{dx}=\left(sin2x\right)y^{1/3}$ satisfying y(0)= 0 is

• A. $y\left(x\right)=\sqrt{\frac{8}{27}}si{n}^{3}x$
• B. $y\left(x\right)=\sqrt{\frac{8}{27}}co{s}^{3}x$
• C. $y\left(x\right)=\sqrt{\frac{27}{8}}si{n}^{3}x$
• D. $y\left(x\right)=\sqrt{\frac{8}{27}}si{n}^{2}x$

Answer 1

The correct option is A $y\left(x\right)=\sqrt{\frac{8}{27}}si{n}^{3}x$

$\frac{dy}{dx}=\left(sin2x\right)y^{1/3}$

$\Rightarrow \left(y^{-1/3}\right)dy=\left(sin2x\right)dx$

Integrating both sides

$\Rightarrow \frac{y^{2/3}}{2/3}=-\frac{cos2x}{2}+C$ ..(i)

$\because$ y(0) = 0

so, $0=-\frac{1}{2}+C\Rightarrow C=\frac{1}{2}$

From equation (i)

$\Rightarrow y^{2/3}=\frac{2}{3}[\frac{-cos2x}{2}+\frac{1}{2}]$

$=\frac{2}{3}\left[\frac{-1+2si{n}^{2}x+1}{2}\right]$
$y^{2/3}=\frac{2}{3}si{n}^{2}x$

$\Rightarrow y={\left(\frac{2}{3}si{n}^{2}x\right)}^{3/2}$
$=\pm \sqrt{\frac{8}{27}}si{n}^{3}x$