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find general solution of following

0 votes
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)
asked Nov 16, 2015 in CALCULUS by anonymous

4 Answers

0 votes

(1a)

The differential equation is image.

Find the general solution using variable separable method.

image.

Rewrite the differential equation separating the variables.

image

Integrate on both sides.

image

image

image

image

image

image

image

image

image

Apply logarithm on each side.

image

image, where image.

answered Nov 16, 2015 by Lucy Mentor
edited Nov 16, 2015 by Lucy
0 votes

(2a)

The differential equation is image.

Rewrite the differential equation.

image

image

The equation is in the form of linear differential equation image.

Solution of first-order linear differential equation is image.

image

image

image

image

image

image.

Solution of the differential equation is image.

answered Nov 16, 2015 by Lucy Mentor
edited Nov 16, 2015 by Lucy
0 votes

(3a)

Step 1:

The differential equation is image.

image

Apply integral on each side.

image.

Consider image.

Apply integration by parts formula: image.

Let image.

Apply derivative on each side.

image.

Let image.

Apply integral on each side.

image

image.

Substitute corresponding values in image.

image

image

image

image.

Step 2:

Consider image.

Let image.

Apply derivative on each side with respect to image.

image

image.

Substitute image and image in image.

image

image

image

image

image

Substitute image.

image

image.

image.

Step 3:

Substitute image and image in image.

image

image

image

image.

The general solution is image.

Solution:

The general solution is image.

answered Nov 16, 2015 by Sammi Mentor
0 votes

(3b)

Step 1:

The differential equation is image.

The equation is in the form of linear differential equation image.

Solution of first order linear differential equation is image.

image

image

image

Step 2:

Consider image.

Apply integration by parts: image.

Let image.

Apply derivative on each side.

image

Let image.

Apply integral on each side.

image

Substitute the corresponding values in integration by parts.

image

image

Substitute image in equation (1).

image

image

image

Solution:

Solution of the differential equation is image

answered Nov 16, 2015 by Lucy Mentor
edited Nov 16, 2015 by Lucy

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