Download App

Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following initial value problems:
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y + x}\sin\Big(\frac{\text{y}}{\text{x}}\Big)=0,\text{y}(2)=\text{x}$
✓

Answer

$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y + x}\sin\Big(\frac{\text{y}}{\text{x}}\Big)=0,\text{y}(2)=\text{x}$
It is a homogeneous equation. put y = vx
and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$
So, $\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{vx}}{\text{x}}-\sin\Big(\frac{\text{vx}}{\text{x}}\Big)$
$\text{x}\frac{\text{dv}}{\text{dx}}=-\sin\text{v}$
$\frac{\text{dv}}{\sin\text{v}}=-\frac{\text{dx}}{\text{x}}$
$\text{cosec(v)dv}=-\frac{\text{dx}}{\text{x}}$
integrating both sides we get,
$\log(\text{cosec(v)}-\cot(\text{v}))=-\log\text{x}+\log\text{c}$
$\log\Big(\text{cosec}\Big(\frac{\text{y}}{\text{x}}\Big)-\cot\Big(\frac{\text{y}}{\text{x}}\Big)\Big)=-\log\text{x}+\log\text{c}$
Putting the values $\text{x}=2$ and $\text{y}=\pi$
$\log\Big(\text{cosec}\Big(\frac{\pi}{2}\Big)-\cot\Big(\frac{\pi}{2}\Big)\Big)=-\log2+\log\text{C}$
$\text{C}=0$
$\log\Big(\text{cosec}\Big(\frac{\text{y}}{\text{x}}\Big)-\cot\Big(\frac{\text{y}}{\text{x}}\Big)\Big)=-\log\text{x}$

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

Solve the following differential equation:$\text{x}\frac{\text{dy}}{\text{dx}}=\text{x}+\text{y}$
→
If $\text{y}=\Big[\log\Big(\text{x}+\sqrt{\text{x}^2+1}\Big)\Big]^2$ show that $(1+\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}=2$
→
If the co-ordinates of the vertices of an equilateral triangle with sides of length ‘a’ are $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3)$, then $\begin{vmatrix}\text{x}_1&\text{y}_1&1\\\text{x}_1&\text{y}_2&1\\\text{x}_2 &\text{y}_3&1\end{vmatrix}^2=\frac{3\text{a}^4}{4}.$
→
Discuss the continuity of the f(x) at the indicated points f(x) = |x - 1| + |x + 1| at x = -1, 1.
→
A bag contains 4 white and 5 black balls and another bag contains 3 white and 4 black balls. A ball is taken out from the first bag and without seeing its colour is put in the second bag. A ball is taken out from the latter. Find the probability that the ball drawn is white.
→
If $\text{A}=\begin{bmatrix}1&0\\0&1\end{bmatrix},\text{B}\begin{bmatrix}1&0\\0&-1\end{bmatrix}$ and $\text{C}=\begin{bmatrix}0&1\\1&0\end{bmatrix},$ then show that $A^2 = B^2 = C^2 = l_2$.
→
Find the vector equations of the following planes in scalar product form $(\vec{\text{r}}\cdot\vec{\text{n}}=\text{d}):$
$\vec{\text{r}}=(2\hat{\text{i}}-\hat{\text{k}})+\lambda\hat{\text{i}}+\mu(\hat{\text{i}}-2\hat{\text{j}}-\hat{\text{k}})$
→
Using differentials, find the approximate values of the following:
$\log_\text{e}10.02$ it being given that $\log_\text{e}10=2.3026$
→
Show that the following triads of vectors are coplanar:
$\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}},\vec{\text{b}}=-2\hat{\text{i}}+3\hat{\text{j}}-4\hat{\text{k}},\vec{\text{c}}=\hat{\text{i}}-3\hat{\text{j}}+5\hat{\text{k}}$
→
Find the shortest distance between the lines
$\vec{\text{r}}=\Big(4\hat{\text{i}}-\hat{\text{j}}\Big)+\lambda\Big(\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\Big)$ $\text{and}\ \vec{\text{r}}=\Big(\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}\Big)+\mu\Big(2\hat{\text{i}}+4\hat{\text{j}}-5\hat{\text{k}}\Big).$
→