solution ofdy/dx=y^3/e^[2x]+y^2.....[] is normal bracket

Hi Praneeth,

This is a typical quesiton, where you need to covert the differential equation to a stadrad form of solvable differential equation:

Take the reciprocals and you have,

dx/dy = (e^2x + y^2)/(y^3)

Make the substitution e^x = t

Now differentiate wrt 'y' on both sides and you have, e^x dx/dy = dt/dy

or dx/dy = (1/t)*dt/dy

Substitue now and you have, (i/t)dt/dy = (t^2 + y^2)/(y^3)

or, dt/dy = (t^3 + t*y^2)/(y^3)

This is a DE in standard form (a homogeneous differential equation, which is solved by the substitution of t = ky)

ie dt/dy = k + y*dk/dy = k^3 + k

ie you have the equation y*dk/dy = k^3, which is variable seperable

ie dk/(k^3) = dy/y......

Now you can proceed and integrate..... and then substitute back for k = (e^x)/y.

That solves the question.....

Hope that helps.

All the Best.

Regards,

Ashwin (IIT Madras)

Approved