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

 

Answers (1)

 

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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