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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)
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