InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
The differential equations , find the particular solution satisfying the given condition:`[xsin^2(y/x)-y]dx+x dy=0; y=pi/4`when x = 1 |
|
Answer» Given that , `sin^(2)((y)/(x))-(y)/(x)+(dy)/(dx)=0` `implies (dy)/(dx)=(y)/(x)-sin^(2)((y)/(x))`……….`(1)` Given differential equation is homogenous. Let `y=vximplies(dy)/(dx)=v+x(dv)/(dx)` From equation `(1)`, `v+x(dv)/(dx)=v-sin^(2)v` `x(dv)/(dx)=-sin^(2)v` `implies cosec^(2)vdv=-(1)/(x)dx` On integration `intcosec^(2)vdv=-int(dx)/(x)implies-cotv=-log|x|+C` `implies log|x|-cotv=Cimplies log|x|-cot((y)/(x))=C`.........`(2)` when `x=1`, then `y=(pi)/(4)implieslog|1|-cot(pi)/(4)=C` `implies C=0-1=-1` put the value of `C` in equation `(2)`, `log|x|-cot((y)/(x))=-1` `implies log|x|-cot((y)/(x)=-loge` `implies cot((y)/(x))=log|ex|` which is the required solution of the given equation. |
|