KVPY Sample Paper KVPY Stream-SX Model Paper-3

If \[y=y(x)\] is the solution of the differential equation, \[x\frac{dy}{dx}+2y={{x}^{2}}\], \[y(1)=1,\] then \[y\left( \frac{1}{2} \right)\] is equal to:

A) \[\frac{7}{64}\]

B) \[\frac{13}{16}\]

C) \[\frac{49}{16}\]

D) \[\frac{1}{4}\]

Correct Answer: C

Solution :

\[\frac{dy}{dx}+\left( \frac{2}{x} \right)\,y\,=\,x\]is a Linerar D.E

If \[{{e}^{\int{2/x/dx}}}={{x}^{2}}\]

\[\therefore \] \[y\times {{x}^{2}}=\,\int{(x\times {{x}^{2}})dx}\]

\[\Rightarrow \] \[y{{x}^{2}}=\,\frac{{{x}^{4}}}{4}+c\,and\,y\,(1)\,=\,1\]

\[\Rightarrow \] \[c\,=\,\frac{3}{4}\]

Put, \[x\,=\,\frac{1}{2}\,\text{then},\]\[y\,=\,\frac{49}{16}.\]