# The solution of the differential equation $x\frac{d^{2}y}{dx^{2}}=1$, given that $y=1, \: \: \frac{dy}{dx}=0, \: when \: x=1$, is Option 1) $y=x\log x+x+2$ Option 2) $y=x\log x-x+2$ Option 3) $y=x\log x+x$ Option 4) $y=x\log x-x$

Solution of Differential Equation -

$\frac{\mathrm{d}y }{\mathrm{d} x} =f\left ( ax+by+c \right )$

put

$Z =ax+by+c$

- wherein

Equation with convert to

$\int \frac{dz}{bf\left ( z \right )+a} =x+c$

$x\frac{d^{2}y}{dx^{2}}=1\\ \frac{d^{2}y}{dx^{2}}=\frac{1}{x}\\$

Integrating both sides

$\frac{dy}{dx}=logx+c\\ \\0=log1+c\Rightarrow\ c=0\\ \therefore \frac{dy}{dx}=logx\\ \int dy=\int logxdx\\ y=xlogx-x+c\\ 1=1.log1-1+c=c-1\\ c=2\\ \therefore y=xlogx-x+2$

Option 1)

$y=x\log x+x+2$

This option is incorrect

Option 2)

$y=x\log x-x+2$

This option is correct

Option 3)

$y=x\log x+x$

This option is incorrect

Option 4)

$y=x\log x-x$

This option is incorrect

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