Download App

Differential Equations — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDifferential Equations2 Marks
Question
Verify that the function $x y=a e^{x}+b e^{-x}+x^{2} ($implicit or explicit$)$ is a solution of the differential equation $x \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}-x y+x^{2}-2=0$
✓

Answer

It is given that $xy = a e^x + b e^{-x} + x^2$
Now, differentiating both sides $w.r.t. x,$ we get,
$\Rightarrow y+x \cdot \frac{d y}{d x}=a \cdot e^{x}-b e^{-x}+2 x ...(i)$
Now, Again differentiating both sides $w.r.t. x$, we get,
$\Rightarrow \frac{d y}{d x}+\frac{d y}{d x}+x \cdot \frac{d^{2} y}{d x^{2}}=a e^{x}+b e^{-x}+2 ...(ii)$
Now, Using Equations. $(i)$ and $(ii),$ we get,
$\text{LHS}\ x \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}-x y+x^{2}-2$
$= a e^{x}+b e^{-x}+2-\left[a e^{x}+b e^{-x}+x^{2}\right]+x^{2}-2$
$= a e^{x}+b e^{x}+2 -a e^{x}-b e^{-x}-x^{2}+x^{2}-2 $
$= 0$
$\Rightarrow \text{LHS = RHS}.$
Therefore, the given function is the solution of the corresponding differential equation.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "2 Marks Questions" from Differential Equations→Differential Equations — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Construct a $2 \times 3$ matrix $A = [a_{ij}]$ whose elements $a_{ij}$ are give by: $a_{ij} = 2i - j$
→
Differentiate $a^x w.r.t. x,$ where a is a positive constant.
→
Write the vector equation of a line given by $\frac{\text{x}-5}{3}=\frac{\text{y}+4}{7}=\frac{\text{z}-6}{2}.$
→
If $\vec{\text{a}}\text{ and }\vec{\text{b}}$ represent two adjacent sides of a parallelogram, then write vectors representing its diagonals.
→
Define differentiability of a function at a point.
→
If $A$ is a square matrix of order $3$ such that $|A| = 2,$ then write the value of adj $($adj $A).$
→
Find x, y, a and b if $\begin{bmatrix}2\text{x}-3\text{y}&\text{a}-\text{b}&3\\1&\text{x}+4\text{y}&3\text{a}+4\text{b}\end{bmatrix}=\begin{bmatrix}1&-2&3\\1&6&29\end{bmatrix}$
→
Find minors and cofactors of all the elements of the determinant $\left|\begin{array}{cc}1 & -2 \\ 4 & 3\end{array}\right|$
→
If A' = $\left[\begin{array}{cc} {-2} & {3} \\ {1} & {2} \end{array}\right] \text { and } \mathbf{B}=\left[\begin{array}{rr} {-1} & {0} \\ {1} & {2} \end{array}\right]$, then find (A + 2B)′
→
Evaluate the following integrals:
$\int\frac{1}{1-\cos2\text{x}}\text{dx}$
→