Mathematics - Differential Equation Question with Solution | TestHub

$y=\frac{1}{8} \int \frac{dx}{\sqrt{4+\sqrt{9+\sqrt{x}} . \sqrt{x} . \sqrt{9+\sqrt{x}}}}$

Put  $\sqrt{9+\sqrt{x}}=t \Rightarrow \frac{dx}{\sqrt{x} . \sqrt{9+\sqrt{x}}}=4dt$

$\therefore y=\frac{4}{8} \int \frac{dt}{\sqrt{4+t}}$

$\Rightarrow y= \sqrt{4+t}+C$

$\Rightarrow y\left(x\right)= \sqrt{4+\sqrt{9+\sqrt{x}}}+C$

At   $x=0:y\left(0\right)= \sqrt{7} \Rightarrow C=0$

$\therefore y\left(x\right)= \sqrt{4+\sqrt{9+ \sqrt{x}}}$

$\Rightarrow y\left(256\right)=3$