Solve the following differential equations by the method of

. Solve the following differential equations by the method of exact differentials:

a) (5x + 4y) + (4x - 8y³) dy/dx = 0,     y(2) =1

Solution

(5x + 4y) + (4x - 8y3) dy/dx = 0,     y(2) =1

we can write

(5x + 4y) + (4x - 8y3) dy/dx = 0,

=> (5x + 4y)dx + (4x - 8y3) dy = 0,   

=> 5xdx + 4ydx + 4xdy - 8y3dy = 0,   

=> 5xdx - 8y3dy + 4ydx + 4xdy = 0,   

=> 5xdx - 8y3dy + 4(ydx + xdy) = 0,

=> 5xdx - 8y3dy + 4d(xy) = 0,

integrating both side we get

5/2 x^2 - 2y^4 + 4xy +c = 0

now y(2) = 1;

(5/2) * 2^2 - 2* 1^4 + 4*2*1 + c = 0;

C = -16

so

5/2 x^2 - 2y^4 + 4xy = 16 is the solution of differential equation.