InterviewSolution
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InterviewSolution
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If `y+d/(dx)(x y)=x(sinx+logx),fin dy(x)dot` |
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Answer» Given differential equation is `" "y+(d)/(dx)(xy)=x(sinx+logx)` `rArr" "y+x(dy)/(dx)+y=x(sinx+logx)` `rArr " "x(dy)/(dx)+2y=x(sinx+logx)` `rArr" "(dy)/(dx)+(2)/(x)y=sinx+logx` which is a linear differential equation. On comparing it with `" "(dy)/(dx)+Py=Q,` we get `" "P=(2)/(x),Q=sinx+logx` `" "IF=e^(int(2)/(x)dx)=e^(2logx)=x^(2)` The general solution is `" "y*x^(2)=int(sinx+logx)x^(2)dx+C` `rArr" "y*x^(2)=int(x^(2)sinx+x^(2)logx)dx+C` `rArr" "y*x^(2)=intx^(2)sinxdx+int x^(2)logxdx+C` `rArr" "y*x^(2)=I_(1)+I_(2)+C" "`...(i) Now, `" "I_(1)=intx^(2)sinxdx` `" "=x^(2)(-cosx)+int2xcosxdx` `" "=-x^(2)cosx+[2x(sinx)-int2sinxdx]` `" "I_(1)=-x^(2)cosx+2xsinx+2cosx" "`...(ii) and `" "I_(2)=intx^(2)logxdx` `" "=logx*(x^(3))/(3)-int(1)/(x)*(x^(3))/(3)dx` `" "=logx*(x^(3))/(3)-(1)/(3)intx^(2)dx` `" "=logx*(x^(3))/(3)-(1)/(3)*(x^(3))/(3)" "`...(iii) On substituting the value of `I_(1) and I_(2)` in Eq. (i), we get `" "y*x^()=-x^(2)cosx+2xsinx+2cosx+(x^(3))/(3)logx-(1)/(9)x^(3) +C` `therefore " "y=-cosx+(2sinx)/(x)+(2cosx)/(x)+(x)/(3)logx-(x)/(9)+Cx^(-2)` |
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