Given the homogeneous equation x^2ydy+xy^2dx=0, use y=ux, u=y/x and dy=udx+xdu to solve the differential equation. solve for y.

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Step-by-step explanation:

Given homogeneous equation

Substitute y=ux ,

Now,

Integrating both side we get

lnu=-2lnx+lnC

Where lnC= integration constant

Cancel ln on both side

Substitute

Then we get

xy=C

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(x3 + 6x2 + 2x - 6) • (x - 1)

step-by-step explanation:

polynomial long division :

3.2     polynomial long division

dividing :   x4+5x3-4x2-8x+6

                              ("dividend")

by         :     x-1     ("divisor")

dividend     x4   +   5x3   -   4x2   -   8x   +   6

- divisor   * x3     x4   -   x3            

remainder         6x3   -   4x2   -   8x   +   6

- divisor   * 6x2         6x3   -   6x2        

remainder             2x2   -   8x   +   6

- divisor   * 2x1             2x2   -   2x    

remainder               -   6x   +   6

- divisor   * -6x0               -   6x   +   6

remainder                     0

quotient :   x3+6x2+2x-6   remainder:   0