InterviewSolution
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InterviewSolution
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Show that the differential equation `(y^2-x^2)dy=3xy dx` is homogenous and solve it |
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Answer» The given differential equation may be written as `(dy)/(dx)=(3xy)/(y^(2)-x^(2))`…………..(i) On dividing the Nr and Dr of RHS of (i) by `x^(2)`, we get `(dy)/(dx)=(3(y/x))/{(y/x)^(2)-1}=f(y/x)`. Thus, the given differential equation is homogenous. Putting `y=vx` and `(dy)/(dx)=v+x(dv)/(dx)` in (i), we get `v+x(dv)/(dx)=(3v)/(v^(2)-1)` `rArr x(dv)/(dx){(3v)/(v^(2)-1)-v}=(4v-v^(3))/(v^(2)-1)` `rArr (v^(2)-1)/(4v-v^(3))dv=1/xdx` `rArr int (v^(2)-1)/(v(2-v)(2+v))dv=1/xdx`...........(ii) Let `(v^(2)-1)/(v(2-v)(2+v))=A/v+B/(2-v)+C/(2+v)`. Then, `(v^(2)-1)-= A(2-v)(2+v)+Bv(2+v)+Cv(2-v)`.............(iii) Putting, `v=0` on each side of (iii), we get `A=-1/4`. Putting v=2 on each side of (iii), we get `B=3/8`. Putting `v=-2` on each side of (iii), we get `C=-3/8`. `therefore (v^(2)-1)/(v(2-v)(2+v))=-1/(4v)+3/(8(2-v))-3/(8(2+v))`................(iv) Putting, these values from (iv) in (ii), we get `-1/4int(dv)/(v)=3/8int(dv)/(2+v)=int1/xdx` `rArr int1/xdx+1/4int(dv)/v+3/8int(-dv)/(2-v)=int1/xdx` `rArr int1/xdx+1/4int(dv)/v+3/8int(-dv)/(2-v)+3/8int(dv)/(2+v)=log|C_(1)|` `rArr 8log|x|+2log|v|+3log|2-v|+3log|2+v|=8log|C_(1)|` `rArr |x^(8)v^(2)(2-v)^(3)(2+v)^(3)|=C_(1)^(8)=C` (say) `rArr |x^(8)v^(2)(2-v)^(3)(2+v)^(3)|=C_(1)^(8)=C` (say) `rArr x^(8)y^(2)/x^(2)(2-y/x)^(3)(2+y/x)^(3)=C` `rArr y^(2)(2x-y)^(3)(2x+y)^(3)=C`, where C is an arbirary constant. `rArr y^(2)(4x^(2)-y^(2))^(2)=C`, which is the required solution. |
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