InterviewSolution
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InterviewSolution
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Show that the differential equation (x^(3)-3xy^(2))dx=(y^(3)-3x^(2)y)dy is homogenous and solve it. |
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Answer» Solution :The given differential EQUATION may be written as `(DY)/(DX)=(x^(3)-3xy^(2))/(y^(3)-3x^(2)y)`………..(i) On dividing the Nr and Dr of RHS of (i) by `x^(3)`, we get `(dy)/(dx)=(1-3(y/x)^(2))/((y/x)^(2)-3(y/x))=f(y/x)`. Thus, the given differential equation is homogenous. Putting, `v=vx` and `(dy)/(dx)=v+x(dv)/(dx)` in (i), we get `(dy)/(dx)=(1-3(y/x)^(2))/((y/x)^(3)-3(y/x))=f(y/x)`. Thus, the given differential equation is homogeneous. Putting, `y=vx` and `(dy)/(dx)=v+x(dv)/(dx)` in (i), we get `=v+x(dv)/(dx)=(x^(3)-3v^(2)x^(3))/(v^(3)x^(3)-3vx^(3))` `rArr v+x(dv)/(dx)=(1-3v^(2))/(v^(3)-3v)` `rArr x(dv)/(dx)=((1-3v^(2))/(v^(3)-3v)-v)` `rArr x(dv)/(dx)=(1-v^(4))/(v^(3)-3v)` `rArr (3v-v^(3))/(v^(4)-1)dv=int(dx)/x`.....................(ii) Let `((3v-v^(3))/(v^(4)-1)=A/(v-1)+B/(v+1)+(Cv+D)/(v^(2)+1))`. Then, `(3v-v^(3))-=A(v+1)(v^(2)+1)+B(v-1)(v^(2)+1)+(Cv+D)(v-1)(v+1))`............(iii) Putting v=1 on each side of (iii), we get `A=1/2`. Putting v`=-1` on each side of (iii), we get `B=1/2`. Comparing coefficients of `v^(3)` on both sides of (iii), we get `A+B+C=-1 rArr 1/2+1/2+C=-1 rArr C=-2`. Comparing the independent terms on both sides of (iii), we get `A-B-D=0 rArr D=(A-B)=(1/2-1/2)=0`. `therefore ((3v-v^(3))/(v^(4)-1)=1/(2(v-1))+1/(2(v+1))-(2v)/(v^(2)+1))`. Putting this value in (ii), we get `1/2int(dv)/(v-1)+1/2int(dv)/(v+1)-int(2v)/(v^(2)+1)dv=int(dx)/x` `rArr 1/2log|v-1|+1/2log|v+1|-LOG|v^(2)+1|=log|x|+log|C|`, where C is an arbitrary constant. `rArr log|v-1|+log|v+1|-2log|v^(2)+1|=2log|x|+2log|C|` `rArr log|((v-1)(v+1))(v^(2)+1)^(2)|=log|C^(2)x^(2)| rArr log|(v^(2)-1)/(v^(2)+1)^(2)|=log|C^(2)x^(2)|` `rArr (y^(2)-x^(2))=C^(2)(y^(2)+x^(2))^(2)[therefore v=y/x]`. Hence, `(y^(2)-x^(2))=C^(2)(y^(2)+x^(2))^(2)` is the required solutions. |
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