InterviewSolution
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InterviewSolution
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Show that the differential equation `((x-y)dy)/(dx)=x+2y ,`ishomogeneous and solve it. |
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Answer» `(x-y)(dy)/(dx)=x+2y` `implies (dy)/(dx)=(x+2y)/(x-y)`………`(1)` Let `f(x,y)=(x+2y)/(x-y)` `implies f(lambdax,lambday)=(lambdax+2lambday)/(lambdax-lambday)=(lambda(x+2y))/(lambda(x-y))` `=(x+2y)/(x-y)=f(x,y)` `:.` Given differential equation is homogenous. Let `y=vx` `implies (dy)/(dx)=v+x(dv)/(dx)` Put this value in equation `(1)` `v+x(dv)/(dx)=(x+2vx)/(x-vx)=(1+2v)/(1-v)` `implies x(dv)/(dx)=(1+2v)/(1-v)-v` `=(1+2v-v+v^(2))/(1-v)=(1+v+v^(2))/(1-v)` `implies (v-1)/(v^(2)+v+1)dv=-(dx)/(x)` `implies int(v-1)/(v^(2)+v+1)dv=-int(dx)/(x)` `implies int((2v+1)-3)/(v^(2)+v+1)dv-3int(1)/(v^(2)+v+1)dv` `=-2logx+c` `impliesl og(v^(2)+v+1)-3int(1)/((v+(1)/(2))^(2)+((sqrt(3))/(2))^(2))dv` `=-logx+c` `implies log(v^(2)+v+1)+logx^(2)` `-3*(1)/(sqrt(3)//2)tan^(-1)"(v+(1)/(2))/(sqrt(3)//2)=c` `implies log(x^(2)v^(2)+vx^(2)+x^(2))-2sqrt(3)tan^(-1)"(2v+1)/(sqrt(3))=c` `implies log(y^(2)+xy+x^(2))-2sqrt(3)tan^(-1)"(2y+x)/(xsqrt(3))=c` |
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