InterviewSolution
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InterviewSolution
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Thedifferential equations , find the particular solution satisfying the givencondition:(x + y) dy + (x y) dx = 0; y = 1 when x =1 |
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Answer» Given that, `(x+y)dy+(x-y)dx=0` `implies (dy)/(dx)=(y-x)/(x+y)`………`(1)` Given differential equation is homogenous. Let `y=vx` `implies (dy)/(dx)=v+x(dv)/(dx)` `implies` From equation `(1)`, `v+x(dv)/(dx)=(vx-x)/(x+vx)` `implies v+x(dv)/(dx)=(x(v-1)/(x(1+v)impliesv+x(dv)/(dx)=(v-1)/(1+v)` `impliesx(dv)/(dx)=(v-1)/(1+v)-v` `impliesx(dv)/(dx)=(v-1-v-v^(2))/(1+v)` `implies -x(dv)/(dx)=(1+v^(2))/(1+v)=((v+1))/(1+v^(2))dv=-(1)/(x)dx` On integration, `int((v+1))/(1+v^(2))dv=-int(dx)/(x)` `impliesint(v)/(1+v^(2))dv+int(1)/(1+v^(2))dv=-int(dx)/(x)` `implies(1)log|v^(2)+1|+tan^(-1)v+log|x|=C` `implies(1)/(2)log((y^(2)+x^(2))/(x^(2)))+log|x|+tan^(-1)((y)/(x))=C` `implies log((y^(2)+x^(2))/(x^(2)))+2log|x|+2tan^(-1)((y)/(x))=2C` `implieslog((y^(2)+x^(2))/(x^(2)))+logx^(2)+2tan^(-1)((y)/(x))=2C` `implies log((y^(2)+x^(2))x^(2))/(x^(2))+2tan^(-1)((y)/(x))=2C` `implies log(x^(2)+y^(2))+2tan^(-1)((y)/(x))=A` (put `2C=A`) .........`(2)` Given, `x=1`, `y=1` `log2+(2xx(pi)/(4))=A` `implies A=(pi)/(2)+log2` put the value of `A` in equation `(2)`, `log(x^(2)+y^(2))+2tan^(-1)((y)/(x))=(pi)/(2)+log2` which is the required solution of the given differential equation |
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