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If y+d/(dx)(xy)=x(sinx+logx), find y(x)....

If `y+d/(dx)(xy)=x(sinx+logx)`, find `y(x)`.

Text Solution

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The given differential equation is
`y+d/(dx)(xy) = x(sinx+logx)`
i.e., `x(dy)/(dx)+2/xy=sinx+logx`.............(1)
This is a linear different equation
I.F.`=e^(2int1/xdx)= e^(2logx)=x^(2)`............(2)
Thus, solution is given by
`yx^(2)=intx^(2)(sinx+logx)dx+c`
`=-x^(2)cosx+2xsinx+2cosx+x^(3)/(3)logx-x^(3)/9+c`
or `y=-cosx+2/xsinx+2/x^(2)cosx+x/3logx-x/3+c/x^(2)`
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Knowledge Check

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