Question

Solve the following differential equation by finding h and k so that the substitutions xequalsuplus​h, yequalsvplusk transform it into the homogeneous equation StartFraction dv Over du EndFraction equals StartFraction u minus v Over u plus v EndFraction dy/dx= (x-y-1)/(x+y+1)

Answer 1

Solve the given the DE,
`(dy)/(dx) =(x-y-1)/(x+y+1)`.

Rearranging terms,

=>
`(dy)/(dx) =(x-y-1)/(x+y+1)`

=>
`(x+y+1)dy=(x-y-1)dx`

=>
`-(x-y-1)dx+(x+y+1)dy=0`

=>
`(-x+y+1)dx+(x+y+1)dy=0`

Checking to see if this is an exact DE,

`M_(y)=1`

`N_(x)=1`

This is an exact DE.

Integrate each M with respect to x and N with respect to y,

=>
`\int\ {-x+y+1} \, dx =-(x^2)/(2) +xy+x`

=>
`\int\ {x+y+1} \, dy =xy+(y^2)/(2)+y`

Thus the solution to the DE is,
`-(x^2)/(2)+(y^2)/(2)+x+y+xy=c`