Question

Solve the Differential Equation

y = 2px + yp²

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Answer 1

Answer :

y² = 2cx + c²

Solution :

Here ,

The given differntial equation is ;

y = 2px + yp² , where p = dy/dx -------(1)

The given differntial equation can be rewritten as ;

=> 2px = y - p²y

=> 2x = y/p - p²y/p

=> 2x = y(1/p - p) ------(2)

Now ,

Differentiating eq-(2) with respect to y , we get ;

=> 2•(dx/dy) = y•(-1/p²-1)•(dp/dy) + (1/p - p)

=> 2/p = -y•(1/p² + 1)•(dp/dy) + 1/p - p

=> y•(1/p² + 1)•(dp/dy) = -2/p + 1/p - p

=> y•[ (1 + p²)/p² ]•(dp/dy) = - 1/p - p

=> y•[ (1 + p²)/p² ]•(dp/dy) = -(1 + p²)/p

=> (y/p)•(dy/dy) = -1

=> dp/p = -dy/y -----(3)

Now ,

Integrating both sides of eq-(3) , we get ;

=> logp = - logy + logc

=> logp = log(c/y)

=> p = c/y

Now ,

Putting p = c/y in eq-(1) , we get ;

=> y = 2cx/y + c²y/y²

=> y = 2cx/y + c²/y

=> y² = 2cx + c²

Hence ,

Required solution is y² = 2cx + c²

Answer 2

Answer:

Step-by-step explanation: