Download App

Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations4 Marks
Question
Solve the following differential equation:
$2\text{xy dx}+(\text{x}^2+2\text{y}^2)\text{dy}=0$
✓

Answer

Here, $2\text{xy dx}+(\text{x}^2+2\text{y}^2)\text{dy}=0$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{2\text{xy}}{\text{x}^2-2\text{y}^2}$
It is a homogeneous equation.
Put y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$
So,
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{2\text{xvx}}{\text{x}^2+2\text{v}^2\text{x}^2}$
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{2\text{v}}{1+2\text{v}^2}$
$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{2\text{v}}{1-2\text{v}^2}-\text{v}$
$=\frac{2\text{v}-\text{v}+2\text{v}^3}{1+2\text{v}^2}$
$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{\text{v}-2\text{v}^3}{1+2\text{v}^2}$
$\int\frac{1+2\text{v}^2}{\text{v}-2\text{v}^3}\text{dv}=\int\frac{\text{dx}}{\text{x}}\ \dots(\text{i})$
$\frac{1+2\text{v}^2}{\text{v}-2\text{v}^3}=\frac{1+2\text{v}^2}{\text{v}(1-2\text{v}^2)}$
$\frac{1+2\text{v}^2}{\text{v}(1-2\text{v}^2)}=\frac{\text{A}}{\text{v}}+\frac{\text{Bv + C}}{1-2\text{v}^2}$
$\frac{1+2\text{v}^2}{\text{v}(1-2\text{v}^2)}=\frac{\text{A}(1-2\text{v}^2)+(\text{Bv + C)}\text{v}}{\text{v}(1-2\text{v}^2)}$
$1+2\text{v}^2=\text{A}-2\text{Av}^2+\text{Bv}^2+\text{Cv}$
$1+2\text{v}^2=\text{v}^2(-2\text{A + B})+\text{Cv + A}$
Comparing the co-efficients of like powers of v,
A = 1
C = 0
-2A + B = 2
-2 + B = 0
B = 4
$\frac{1+2\text{v}^2}{\text{v}-2\text{v}^3}=\frac{1}{\text{v}}+\frac{4\text{v}}{1-2\text{v}^2}$ 
$\frac{1+2\text{v}^2}{\text{v}-2\text{v}^3}=\frac{1}{\text{v}}-\frac{(-4\text{v})}{(1-2\text{v}^2)}$

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

$\begin{vmatrix}-\text{a}(\text{b}^2+\text{c}^2-\text{a}^2)&2\text{b}^3&2\text{c}^3\\2\text{a}^3&-\text{b}(\text{c}^2+\text{a}^2-\text{b}^2)&2\text{c}^3\\2\text{a}^3&2\text{b}^3&-\text{c}(\text{a}^2+\text{b}^2-\text{c}^2)\end{vmatrix}$
$=\text{abc}(\text{a}^2+\text{b}^2+\text{c}^2)$
→
If $\text{y}=(\cos\text{x})^{\cos\text{x}^{\cos\text{x}^{.....\infty}}},$ prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}^2\tan\text{x}}{(1-\text{y}\log\cos\text{x})}$
→
Evaluate the following integrals:
$\int\frac{\text{x}\sin^{-1}\text{x}^2}{\sqrt{1-\text{x}^4}}\text{ dx}$
→
If $\text{x}=\text{a}(\theta+\sin\theta)\ \text{and}\ \text{y}=\text{a}(1+\cos\theta)$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\frac{\text{a}}{\text{y}^2}.$
→
Show that the points A (1, -2, -8), B (5, 0, -2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.
→
If $\int\limits_{0}^{\text{k}}\frac{1}{2+8\text{x}^2}\text{ dx}=\frac{\pi}{16},$ find the value of k.
→
Show that the following systems of linear equations has infinite number of solutions and solve:
x + y - z = 0,
x - 2y + z = 0,
3x + 6y - 5z = 0
→
Differentiate $\sin^{-1}\Big(4\text{x}\sqrt{1-4\text{x}^2}\Big)$ with respect to $\sqrt{1-4\text{x}^2},$ if:
$\text{x}\in\Big(-\frac{1}{2\sqrt{2}},\frac{1}{\sqrt{2\sqrt{2}}}\Big)$
→
By using the properties of definite integrals, evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x d x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}$
→
Differentiate $\frac{\text{x}}{\sin\text{x}}$ w.r.t. $\sin\text{x}.$
→