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

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

Grade:12

1 Answers

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

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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