1.

Solve the differential equation `x(dy)/(dx)+y=xcosx+sinx ,`given `y(pi/2)=1`

Answer» `dy/dx+y/x=cosx+sinx/x`
`dy/dx+yP(x)=q(x)`
`P(x)=1/x`
`q(x)=cosx+sinx/x`
IF=`e^(intPdx)`
IF=x
`y*IF=intq(x)IFdx+C`
`yx=int(cosx+sinx/x)x+C`
`=xsinx+C`
`xy=xsinx+C`
`x=pi/2`
`y=1`
`C=0`
`xy=xsinx`
`y=sinx`.


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