Find general solution of following separable differential equations,

Question

a) (dy/dx) = (xe^(2-y^2))/(y sqrt(x-2))

b) (dy/dx) +12x^3 y= 48x^3

Answer 1

(a)

Step 1:

The differential equation is .

Rewrite the equation.

.

Apply integration on each side.

Consider the integral .

Solve the integral using substitution method.

Consider .

Apply derivative on each side with respect to .

Substitute and in .

.

Apply formula : .

Substitute .

.

Step 2:

Consider .

Solve the integral using substitution method.

Consider .

Apply derivative on each side with respect to .

 

If then .

Substitute , and in .

Substitute .

.

Step 3:

The integral is .

Substitute the above results in the integral.

.

Therefore, the general solution is .

Solution:

The general solution is .

Answer 2

(b)

The differential equation is .

Rewrite the equation.

.

Apply integral on each side.

Apply formula : .

Apply power rule of integration : .

.

Therefore, the general solution is .

Solution:

The general solution is .