Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS4 Marks
Question
Solve the following initial value problems:
$\frac{\text{dy}}{\text{dx}}-3\text{y}\cot\text{x}=\sin2\text{x},\text{ y}=2,\text{ when x}=\frac{\pi}{2}$
✓

Answer

We have,
$\frac{\text{dy}}{\text{dx}}-3\text{y}\cot\text{x}=\sin2\text{x}\ ....(1)$
Clearly, it is a linear differential equation of the form
$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
Where $\text{P}=-3\cot\text{x}$ and $\text{Q}=\sin2\text{x}$
$\therefore\text{ I.F.}=\text{e}^{\int\text{Pdx}}$
$=\text{e}^{-3\int\cot\text{x dx}}$
$=\text{e}^{-3\log|\sin\text{x}|}$
$=\text{cosec}^3\text{x}$
Multiplying both sides of (1) by $\text{I.F.}=\text{cosec}^3\text{x},$ we get
$\text{cosec}^3\text{x}\Big(\frac{\text{dy}}{\text{dx}}-3\text{y}\cot\text{x}\Big)=\sin2\text{x}(\text{cosec}^3\text{x})$
$\Rightarrow\text{cosec}^3\text{x}\Big(\frac{\text{dy}}{\text{dx}}-3\text{y}\cot\text{x}\Big)=2\cot\text{x cosec x}$
Integrating both sides with respect to x, we get
$\text{y }\text{cosec}^3\text{x}=2\int\cot\text{x}\text{ cosec}\text{ x dx}+\text{C}$
$\Rightarrow\text{y }\text{cosec}^3\text{x}=-2\text{cosec}\text{ x}+\text{C}$
$\Rightarrow\text{y}=-2\sin^2\text{x}+\text{C}\sin^3\text{x}\ ...(2)$
Now,
$\text{y}\Big(\frac{\pi}{2}\Big)=2$
$\therefore\ 2=-2\sin^2\frac{\pi}{2}+\text{C}\sin^3\frac{\pi}{2}$
$\Rightarrow\text{C}=4$
Putting the value of C in (2), we get
$\text{y}=-2\sin^2\text{x}+4\sin^3\text{x}$
$\Rightarrow\text{y}=4\sin^3\text{x}-2\sin^2\text{x}$
Hence, $\text{y}=4\sin^3\text{x}-2\sin^2\text{x}$ is the required solution.

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 "4 Marks" 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

Evaluate the integral $\int_{0}^{1} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x$ using substitution.
→
Five defective mangoes are acciedently mixed with 15 good ones. Four mangoes are drawn at random from this lot. Find the probability distribution of the number of defective mangoes.
→
Evaluate the following integrals:
$\int\limits^{\text{a}}_{-\text{a}}\sqrt{\frac{\text{a}-\text{x}}{\text{a}+\text{x}}}\text{ dx}$
→
A firm manufactures two types of products A and B and sells them at a profit of Rs 2 on type A and Rs 3 on type B. Each product is processed on two machines M1 and M2. Type A requires one minute of processing time on M1 and two minutes of M2; type B requires one minute on M1 and one minute on M2. The machine M1 is available for not more than 6 hours 40 minutes while machine M2 is available for 10 hours during any working day. Formulate the problem as a LPP.
→
If $\text{A}=\begin{bmatrix}2&3\\1&2\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix},$ then find $\lambda,\mu$ so that $\text{A}^2=\lambda\text{A}+\mu\text{I}$
→
show that the differential equation of which $\text{y}=2(\text{x}^2-1)+\text{ce}^{-\text{x}^{2}}$ is a solution, is $\frac{\text{dy}}{\text{dx}}+2\text{xy}=4\text{x}^3$
→
Solve the following L.P.P. graphically:
Maximise Z = 20x + 10y
Subject to the following constraints x + 2y $\leq$ 28,
3x + y $\leq$ 24,
x $\geq$ 2,
x, y $\geq$ 0
→
If y(x) is a solution of the different equation $\Big(\frac{2+\sin\text{x}}{1+\text{y}}\Big)\frac{\text{dy}}{\text{dx}}=-\cos\text{x}$ and $\text{y}(0)=1,$ then find the value of $\text{y}\Big(\frac{\pi}{2}\Big).$
→
Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}\frac{1}{\text{a}}&\text{a}^2&\text{bc}\\\frac{1}{\text{b}}&\text{b}^2&\text{ac}\\\frac{1}{\text{c}}&\text{c}^2&\text{ab} \end{vmatrix}$
→
If $\text{P}(\text{x})=\begin{bmatrix}\cos\text{x}&\sin\text{x}\\-\sin\text{x}&\cos\text{x}\end{bmatrix},$ then show that P(x).P(y) = P(x + y) = P(y)P(x).
→