Question

Find general solution of following separable differential equation

a) (dy/dx) = (xe^(y^2-1)) / (y(x-5))

Find general solution of following linear differential equation

a) x^2(dy/dx) + 3xy = 12x^8

Find general solution of following differential equations

a) y(dy/dx) = (12x^2) / (e^(2-x^3)lny)

b) (dy/dx) + (y/x) = 24e^(2x+1)

Answer 1

(1a)

The differential equation is .

Find the general solution using variable separable method.

.

Rewrite the differential equation separating the variables.

Integrate on both sides.

Apply logarithm on each side.

, where .

Answer 2

(2a)

The differential equation is .

Rewrite the differential equation.

The equation is in the form of linear differential equation .

Solution of first-order linear differential equation is .

.

Solution of the differential equation is .

Answer 3

(3a)

Step 1:

The differential equation is .

Apply integral on each side.

.

Consider .

Apply integration by parts formula: .

Let .

Apply derivative on each side.

.

Let .

Apply integral on each side.

.

Substitute corresponding values in .

.

Step 2:

Consider .

Let .

Apply derivative on each side with respect to .

.

Substitute  and  in .

Substitute .

.

.

Step 3:

Substitute  and  in .

.

The general solution is .

Solution:

The general solution is .

Answer 4

(3b)

Step 1:

The differential equation is .

The equation is in the form of linear differential equation .

Solution of first order linear differential equation is .

Step 2:

Consider .

Apply integration by parts: .

Let .

Apply derivative on each side.

Let .

Apply integral on each side.

Substitute the corresponding values in integration by parts.

Substitute in equation (1).

Solution:

Solution of the differential equation is