Question

If $\mathrm{y}\left(\mathrm{x}\right)$ is solution of $\mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)+2\mathrm{y}={\mathrm{x}}^2,\mathrm{y}\left(1\right)=1,$ then value of $\mathrm{y}\left(\frac{1}{2}\right)$

• A. $-\frac{49}{16}$
• B. $\frac{49}{16}$
• C. $\frac{45}{8}$
• D. $-\frac{45}{8}$

Answer 1

The correct option is B

$\frac{49}{16}$

Explanation for the correct option:

Finding the value of $\mathrm{y}\left(\frac{1}{2}\right):$

The given differential equation is $\mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)+2\mathrm{y}={\mathrm{x}}^2$

Divide by $x$ on both side..

$\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)+\frac{2}{\mathrm{x}}\left(\mathrm{y}\right)=\mathrm{x}$

$\begin{array}{rcl}\text{I.F.}& =& {\mathrm{e}}^{\int \frac{2}{\mathrm{x}}\mathrm{dx}}\\ & =& {\mathrm{e}}^{2\mathrm{lnx}}\\ & =& {\mathrm{x}}^2\end{array}$

The differential equation's solution is

$$\begin{gathered} \mathrm{y}·\left(\mathrm{I}.\mathrm{F}\right)=\int \mathrm{x}\cdot \left(\mathrm{I}.\mathrm{F}\right)\mathrm{dx}+\mathrm{C} \\ \Rightarrow \mathrm{y}\cdot {\mathrm{x}}^2=\int \mathrm{x}\cdot {\mathrm{x}}^2\mathrm{dx}+\mathrm{C} \\ \Rightarrow {\mathrm{yx}}^2=\frac{{\mathrm{x}}^4}{4}+\mathrm{C}...\mathbf{[}\mathbf{\because }\mathbf{}\mathbf{\int }{\mathbf{x}}^{\mathbf{n}}\mathbf{dx}\mathbf{=}\frac{{\mathbf{x}}^{\mathbf{n}\mathbf{+}\mathbf{1}}}{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{+}\mathbf{C}\mathbf{]} \\ \Rightarrow \mathrm{y}=\frac{{\mathrm{x}}^2}{4}+\frac{\mathrm{C}}{{\mathrm{x}}^2}...\left(1\right) \end{gathered}$$

The given equation is $\mathrm{y}\left(1\right)=1$

$\begin{array}{l}1=\frac{1}{4}+\mathrm{C}...\mathbf{[}\mathbf{from}\mathbf{}\mathbf{1}\mathbf{]}\\ \mathrm{C}=\frac{3}{4}\end{array}$

Now, substitute the value of $\mathrm{C}$ in equation $\left(1\right)$, we get

$\begin{array}{l}\mathrm{y}\left(\mathrm{x}\right)=\frac{{\mathrm{x}}^2}{4}+\frac{3}{4{\mathrm{x}}^2}\end{array}$

Substitute $x=\frac{1}{2}$ in the above equation,

$\begin{array}{l}\mathrm{y}\left(\frac{1}{2}\right)=\frac{1}{16}+\frac{3}{4\left(\frac{1}{4}\right)}\\ \Rightarrow \mathrm{y}\left(\frac{1}{2}\right)=\frac{49}{16}\end{array}$

Hence, the correct option is B.