1.

Solve the differential equation `(x+1)(dy)/(dx) =2xy, " given that " y(2)=3.`

Answer» Correct Answer - `y(x+1)^(2)=27e^((2x-4))`
`int (1)/(y)dy =int (2x)/((x+1))dx = 2int ((x+1)-1)/((x+1))dx =2int {1-(1)/((x+1))}dx`
` rArr log |y|=2x - 2 log |x+1|+log |C_(1)|`
`rArr log |y|=log |e^(2x)|-log|(x+1)^(2)|+log |C_(1)|`
`rArr log|(y(x+1)^(2))/(e^(2x))|=log|C_(1)|rArr (y(x+1)^(2))/(e^(2x))=pm C_(1)=C` (say).
Then, `y(x+1)^(2)=Ce^(2x).`
Putting x =2 and y =3 in (i), we get `C=27e^(-4)`.
`therefore y(x+1)^(2)=27e^(2x-4)` is the required solution.


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