InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
show that the given differential equation is homogeneous and solve each of them `(x^2+xy)dy=(x^2+y^2)dx` |
|
Answer» `(dy)/(dx)=(x^(2)+y^(2))/(x^(2)+xy)` `implies v+x(dv)/(dx)=(x^(2)+x^(2)v^(2))/(x^(2)+x^(2)v)` Let `y=vx` `implies (dv)/(dx)=v+x(dv)/(dx)` `implies x(dv)/(dx)=(1+v^(2))/(1+v)-v=(1-v)/(1+v)` `implies (1+v)/(1-v)dv=(dx)/(x)` `implies int((2)/(1-v)-1)dv=int(dx)/(x)+logc` `implies -2log(1-v)-v=logx+logc` `implies -v=logcx+2log(1-v)=log{cx(1-v)^(2)}` `implies cx(1-v)^(2)=e^(-v)` `implies cx(1-(y)/(x))^(2)=e^(-y//x)` `implies c(x-y)^(2)=x*e^(-y//x)` `implies c(x-y)^(2)*e^(y//x)=x` |
|