Question

If the graph of y=f(x) passes through the point (0,1), and dy/dx=xsin(x^2)/y, then f(x)= ?

Answer 1

Step-by-step explanation:

We have,

$\frac{dy}{dx} = \frac{x \sin( {x}^{2} ) }{y}$

$ydy = x \sin( {x}^{2} ) dx$

$\implies \: \int \: ydy = \int x \sin( {x}^{2} ) dx$

$\implies \: \frac{ {y}^{2} }{2} = \frac{1}{2} \int2 x \sin( {x}^{2} ) dx$

$\implies \: \frac{ {y}^{2} }{2} = - \frac{1}{2} \cos( {x}^{2} ) + C$

We have, for x= 0, y=1

So,

$\implies \: \frac{ 1}{2} = - \frac{1}{2} \cos( 0) + C$

$\implies \: \frac{ 1}{4} =C$

So, required solution,

$\implies \: \frac{ {y}^{2} }{2} = - \frac{1}{2} \cos( {x}^{2} ) + \frac{1}{4}$

$\implies \: 2{y}^{2} = - 2 \cos( {x}^{2} ) + 1$