Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
Solve the following differential equation:
$(\sin\text{x})\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}=2\sin^2\text{x}\cos\text{x}$
✓

Answer

We have,
$(\sin\text{x})\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}=2\sin^2\text{x}\cos\text{x}$
$\frac{\text{dy}}{\text{dx}}+\text{y}\cot\text{x}=2\sin\text{x}\cos\text{x}\ \dots(1)$
Clearly, it is a linear differential equation of the form
$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
where
$\text{P}=\cot\text{x}$
$\text{Q}=2\sin\text{x}\cos\text{x}$
$\therefore$ I.F. $=\text{e}^{\int\text{Pdx}}$
$=\text{e}^{\int\cot\text{xdx}}$
$=\text{e}^{\log|\sin\text{x}|}=\sin\text{x}$
Multiplying both sides of (1) by $\sin\text{x},$ we get
$\sin\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}\cot\text{x}\Big)=\sin\text{x}\times2\sin\text{x}\cos\text{x}$
$\sin\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}=2\sin^2\text{x}\cos\text{x}$
Integrating both sides with respect to x, we get
$\text{y}\sin\text{x}=2\int\sin^2\text{x}\cos\text{x dx + C}\ \dots(2)$
Putting $\sin\text{x}=\text{t}$
$\Rightarrow\ \cos\text{x dx = dt}$
Therefore, (2) becomes
$\text{y}\sin\text{x}=2\int\text{t}^2\text{dt + C}$
$\Rightarrow \text{y}\sin\text{x}=\frac{2}3\text{t}^3+\text{C}$
$\Rightarrow \text{y}\sin\text{x}=\frac{2}3\sin^3\text{x}+\text{C}$
Hence, $\text{y}\sin\text{x}=\frac{2}3\sin^3\text{x}+\text{C}$ 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 "5 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

Find the equation of the plane that bisects the line segment joining the points (1, 2, 3) and (3, 4, 5) and is at right angle to it.
→
Show that the height of a closed right circular cylinder of given surface and maximum volume, is equal to the diameter of its base.
→
For the following pairs of matrices verify that $(AB)^{-1} = B^{-1} A^{-1}:$
$\text{A}=\begin{bmatrix}2 & 1 \\5 & 3 \end{bmatrix}\text{ and B}=\begin{bmatrix}4 & 5 \\3 & 4 \end{bmatrix}$
→
Let X denot the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that
$\text{P}(\text{X = x})=\begin{cases}\text{kx},&\text{if}\text{ x}=0\text{ or }1\\2\text{kx},&\text{if x = 2}\\\text{k}(5-\text{x}),&\text{if x = 3 or 4}\\0,&\text{if x > 4}\end{cases}$
where k is a positive constant. Find the value of k. Also find the probability that you will get addmission in
  1. Exactly one college.
  2. At most two colleges.
  3. At least two colleges.
→
Find the area of the region bounded by $\text{y}=\sqrt{\text{x}}$ and $y = x.$
→
Evaluate the following integrals:
$\int\frac{\tan\text{x}}{\sqrt{\cos\text{x}}}\text{dx}$
→
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 2x^3 + 9x^2 + 12x + 20$
→
Find the equation of a curve passing through the point (0, 0) and whose differential equation is $\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}}\sin\text{x.}$
→
If $f(x) = x^3 + 7x^2 + 8x - 9$, find $f(4)$.
→
Using vector method, prove that the point is collinear:
A(-3, -2, -5), B(1, 2, 3) and C(3, 4, 7)
→