Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
For each of the differential equations given in find a particular solution satisfying the given condition:
$\frac{\text{dy}}{\text{dx}}-3\text{y}\cot\text{x}=\sin 2\text{x};\ \text{y}=2\ \text{when x}=\frac{\pi}{2}$
✓

Answer

The given differential equation is $\frac{\text{dy}}{\text{dx}}-3\text{y}\cot\text{x}=\sin\ 2\text{x}.$This is a linear differential equation of the form:
$\frac{\text{dy}}{\text{dx}}+\text{py}=\text{Q}\ (\text{where p}=-3\cot\text{x}\ \text{and}\ \text{Q}=\sin2\text{x})$
$\text{Now, I.F}=\text{e}^{\int\text{pdx}}=\text{e}^{-3\int\cot\text{x}\ \text{dx}}=\text{e}^{-3\log|\sin\text{x}|}=\text{e}^{\log|\frac{1}{\sin^3\text{x}}|}=\frac{1}{\sin^3\text{x}}.$
The general solution of the given differential equation is given by the relation,
$\text{y(I.F)}=\int(\text{Q}\times\text{I.F.})\text{dx}+\text{C}$
$\Rightarrow​​\text{y}\cdot\frac{1}{\sin^3\text{x}}=\int\Big[\sin2\text{x}.\frac{1}{\sin^3\text{x}}\text{dx}+\text{C}\Big]$
$\Rightarrow\text{y cosec}^3\ \text{x}=2\int(\cot\text{x}\ \text{cosec}\ \text{x})\text{dx}+\text{C}$
$\Rightarrow\text{y cosec}^3\ \text{x}=-2\text{cosec}\ \text{x}+\text{C}$
$\Rightarrow\text{y}=-\frac{2}{\text{cosec}^2\text{x}}+\frac{\text{C}}{\text{cosec}^3\text{x}}$
$\Rightarrow\text{y}=-2\sin^2\text{x}+\text{C}\sin^3\text{x}\ \ ...(1)$
$\text{Now,}\ \text{y}=2\ \text{at}\ \text{x}=\frac{\pi}{2}.$
Therefore, we get:
$2=-2+\text{C}$
$\Rightarrow\text{C}=4$
Substituting C = 4 in equation (1), we get:
$\text{y}=-2\sin^2\text{x}+4\sin^3\text{x}$
$\Rightarrow\text{y}=4\sin^3\text{x}-2\sin^2\text{x}$
This is the required particular solution of the given 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 "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

Using properties of determinants, prove that
$\begin{vmatrix} \text{a}^2+2\text{a}& 2\text{a}+1 & 1\$0.3em] 2\text{a}+1 & \text{a}+2 & 1 \$0.3em] 3 & 3 & 1 \end{vmatrix}=(\text{a}-1)^3$
→
Arun and Tarun appeared for an interview for two vacancies. The probability of Arun's selection is $\frac{1}{4}$ and that to Tarun's rejection is $\frac{2}{3}$. Find the probability that at least one of them will be selected.
→
Of all the closed cylindrical cans (right circular), which enclose a given volume of $100\ cm^3,$ which has the minimum surface area ?
→
A bag contains 3 white and 2 black balls and another bag contains 2 white and 4 black balls. One bag is chosen at random. From the selected bag, one ball is drawn. Find the probability that the ball drawn is white.
→
If $\sin(\text{xy})+\frac{\text{y}}{\text{x}}=\text{x}^2-\text{y}^2,$ find $\frac{\text{dy}}{\text{dx}}$
→
Solve the following LPP graphically:
Minimise Z = 3x + 9y
subject to the constraints
x + 3y $\leq$ 60
x + y $\geq$ 10
x $\leq$ y
x $\geq$ 0, y $\geq$ 0.
→
The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.
→
Solve the following systems of homogeneous linear equations by matrix method:
$x + y - z = 0$
$x - 2y + z = 0$
$3x + 6y - 5z = 0$
→
Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{y}=\text{x}^2+4\text{x}+1\text{ at }\text{x}=3$
→
Determine if f defined by:
$\text{f(x)}=\begin{cases}\text{x}^{2} \sin\frac{1}{\text{x}}, \text{if} \ \text{x}\neq0\\0, \ \ \ \ \ \ \ \ \ \ \ \text{if}\ \text{x} = 0\end{cases}$
→