Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS4 Marks
Question
Solve the following initial value problems:
$\frac{\text{dy}}{\text{dx}}+\text{y}\cot\text{x}=2\cos\text{x},\text{ y}\Big(\frac{\pi}{2}\Big)=0$
✓

Answer

We have,
$\frac{\text{dy}}{\text{dx}}+\text{y}\cot\text{x}=2\cos\text{x},\text{ y}\Big(\frac{\pi}{2}\Big)=0$
$\frac{\text{dy}}{\text{dx}}+\text{y}\cot\text{x}=2\cos\text{x}\ ...(1)$
Clearly, it is a linear differential equation of the form
$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
Where $\text{P}=\cot\text{x}$ and $\text{Q}=2\cos\text{x}$
$\therefore\text{ I.F.}=\text{e}^{\int\text{Pdx}}$
$=\text{e}^{\int\cot\text{x dx}}$
$=\text{e}^{\log\sin\text{x}}$
$=\sin\text{x}$
Multiplying both sides of (1) by $\text{I.F.}=\sin\text{x},$ we get
$\sin\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}\cot\text{x}\Big)=2\sin\text{x }\cos\text{x}$
$\Rightarrow\sin\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}=\sin2\text{x}$
Integrating both sides with respect to x, we get
$\text{y}\sin\text{x}=\int\sin2\text{x dx}+\text{C}$
$\Rightarrow\text{y}\sin\text{x}=-\frac{\cos2\text{x}}{2}+\text{C}\ ...(2)$
Now,
$\text{y}\Big(\frac{\pi}{2}\Big)=0$
$\therefore\ 0\times\sin\frac{\pi}{2}=-\frac{\cos\pi}{2}+\text{C}$
$\Rightarrow\text{C}=-\frac{1}{2}$
Putting the value of C in (2), we get
$\text{y}\sin\text{x}=-\frac{\cos2\text{x}}{2}-\frac{1}{2}$
$\Rightarrow2\text{y}\sin\text{x}=-(1+\cos2\text{x})$
$\Rightarrow2\text{y}\sin\text{x}=-2\cos^2\text{x}$
$\Rightarrow\text{y}=-\cot\text{x}\cos\text{x}$
Hence, $\text{y}=-\cot\text{x}\cos\text{x}$ 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

Solve the following systems of linear equations by cramer's rule:
2x + 3y = 10,
x + 6y = 4
→
Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}\sin^2\text{A}&\cot\text{A}&1\\\sin^2\text{B}&\cot\text{B}&1\\\sin^2\text{C}&\cot\text{C}&1\end{vmatrix}$
→
If $\tan^{-1}\bigg(\frac{\text{x} - 2}{\text{x} - 4}\bigg) + \tan^{-1}\bigg(\frac{\text{x} + 2 }{\text{x} + 4 }\bigg) = \frac{\pi}{4} , $find the value of x.
→
Show that the function $\text{f} : \text{R}\rightarrow\text{R}$ defined by $\text{f(x})=\frac{\text{x}}{\text{x}^2+1},\forall\text{ x}\in\text{R}$  is neither one-one nor onto. Also, if $\text{g} : \text{R}\rightarrow\text{R}$ is defined as g(x) = 2x – 1, find fog(x).
→
Using integration, find the area of the following region:
 $\Big\{(\text{x},\text{y}):\frac{\text{x}^2}{9}+\frac{\text{y}^2}{4}\leq1\leq\frac{\text{x}}{3}+\frac{\text{y}}{2}\Big\}$
→
Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{e}^{\theta}\Big(\theta+\frac{1}{\theta}\Big)\text{ and y}=\text{e}^{-\theta}\Big(\theta-\frac{1}{\theta}\Big)$
→
Differentiate the following functions with respect to x:
$\text{x}^{\text{x}^2-3}+(\text{x}-3)^{\text{x}^2}$
→
Let A = {1, 2, 3, ... 9} and R be the relation in A × A defined by (a, b)R(c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)].
→
A small manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry, then sent to the machine shop for finishing. The number of man-hours of labour required in each shop for the production of each unit of A and B, and the number of man-hours the firm has available per week are as follows:
Gadget
Fondry
Machine-shop
A
B
10
6
5
4
Firm's capacity per week
1000
600
The profit on the sale of A is Rs. 30 per unit as compared with Rs. 20 per unit of B. The problem is to determine the weekly production of gadgets A and B, so that the total profit is maximized. Formulate this problem as a LPP.
→
The height of a cone increases by k%, its semi-vertical angle remaining the same. What is the approximate percentage increase
  1. In total surface area, and
  2. In the volume, assuming that k is small?
→