Download App

Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations2 Marks
Question
Verify that the function $y=x \sin 3 x$ (implicit or explicit) is a solution of the differential equation $\frac{d^{2} y}{d x^{2}}+9 y-6 \cos 3 x=0$
✓

Answer

It is given that y = x sin 3x
Now, differentiating both sides w.r.t. x , we get,
$\frac{d y}{d x}=\frac{d}{d x}(x \sin 3 x)=\sin 3 x+x . \cos 3 x .3$
$\Rightarrow \frac{d y}{d x}=\sin 3 x+3 x \cos 3 x$
Now, again differentiating both sides w.r.t. x, we get,
$\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}(\sin 3 x)+3 \frac{d}{d x}(x \cos 3 x)$
$\Rightarrow \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=3 \mathrm{cos} 3 \mathrm{x}+3[\cos 3 \mathrm{x}+\mathrm{x}(-\sin 3 \mathrm{x}) \cdot 3]$
$\Rightarrow \frac{d^{2} y}{d x^{2}}=6 \cos 3 x-9 x \sin 3 x$
Now, substituting the value of $\frac{d^{2} y}{d x^{2}}$ in the LHS of the given differential equation, we get,
$\frac{d^{2} y}{d x^{2}}+9 y-6 \cos 3 x$
= (6.cos 3x - 9x sin 3x) + 9x sin 3x - 6 cos 3x
= 0 = 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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Prove the following results:
$\tan^{-1}\frac{1}{7}+\tan^{-1}\frac{1}{13}=\tan^{-1}\frac{2}{9}$
→
Find the vector equation of a plane passing throught a point with position $2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and perpendicular to the vector $4\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}$
→
Fatima and John appear in an interview for two vacancies for the same post. The probability of Fatima's selection is $\frac{1}{7}$ and that of John's selection is $\frac{1}{5}$. What is the probability that,
Both of them will be selected?
→
If -1 < x < 0, then write the value of $\sin^{-1}\Big(\frac{2\text{x}}{1+\text{cx}^2}\Big)+\cos^{-1}\Big(\frac{1-\text{x}^2}{1+\text{x}^2}\Big).$
→
Write the identity element for the binary operation * on the set $R_0$ of all non-zero real numbers by the rule $\text{a}\times\text{b}=\frac{\text{ab}}{2}$ for all $a, b \in R_0$.
→
Verify that the function $y=e^{x}(a \cos x+b \sin x)$ (implicit or explicit) is a solution of the differential equation $\frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+2 y=0$
→
Let $'o'$ be a binary operation on the set $Q_0$ of all non-zero rational numbers defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{2}$ for all $\text{a},\text{b}\in\text{Q}_0.$
Show that $'o'$ is both commutative and associate.
→
Show that
$(\vec{a}-\vec{b})\times(\vec{a}+\vec{b})=2(\vec{a}\times{\vec{b}})$
→
Write the coordinate axis to which the line $\frac{\text{x}-2}{3}=\frac{\text{y}+1}{4}=\frac{\text{z}-1}{0}$ is perpendicular.
→
Find the set values of $\text{cosec}^{-1}\Big(\frac{\sqrt3}{2}\Big)$
→