Question

solve dy/dx-1/x y=3 x e^x^3 y^2

Answer 1

Answer:

Step-by-step explanation:

Write the equation as (x-y-1)dx - (x+y+3)dy =0 . Then take

x = U-1 , y = V-2 . The equation becomes

( U - V )dU - ( U + V )dV =0 which is homogeneous . Then let V = UW

and obtain ( W^2 + 2W - 1 )dU + U( W + 1 )dW =0 . Separate as

[( W + 1 )/(W-W1)(W-W2)]dW = - dU/U , where W1,2 = -1 +-(2)^1/2.Then

-dU/U = (2^(-3/2))[ (1+W)/(W-W1) - (1+W)/(W-W2) ] . Integrating obtain

(2^(-3/2))[ (1+W1)ln(W-W1) - (1+W2)ln(W-W2) ] = - lnU + lnC. Follows

ln[(U^2)(W-W1)(W-W2)] = lnK , K=C^2. Then

(U^2)(W^2 + 2W -1) = K. Recalling that W = V/U = (y+2)/(x+1) obtain the

solution in term of original variables.