Solve the following differential equations: 2xy dy dx =x^2+y^2 - Vidyadip

Question

Solve the following differential equations:
$2\text{xy}\frac{\text{dy}}{\text{dx}}=\text{x}^2+\text{y}^2$

✓

Answer

Here, $2\text{xy}\frac{\text{dy}}{\text{dx}}=\text{x}^2+\text{y}^2$
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x}^2+\text{y}^2}{2\text{xy}}$
It is homogeneous equation.
Put y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$
So,
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{x}^2+\text{v}^2\text{x}^2}{2\text{xvx}}$
$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{1+\text{v}^2}{2\text{v}}-\text{v}$
$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{1+\text{v}^2-2\text{v}^2}{2\text{v}}$
$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{1-\text{v}^2}{2\text{v}}$
$\frac{2\text{v}}{1-\text{v}^2}\text{dv}=\frac{\text{dx}}{\text{x}}$
$\int\frac{-2\text{v}}{1-\text{v}^2}\text{dv}=-\int\frac{\text{dx}}{\text{x}}$
$\log\big|1-\text{v}^2\big|=-\log|\text{x}|+\log|\text{C}|$
$(1-\text{v}^2)=\frac{\text{C}}{\text{x}}$
$\text{x}\Big(1-\frac{\text{y}^2}{\text{x}^2}\Big)=\text{C}$
$\frac{\text{x}(\text{x}^2-\text{y}^2)}{\text{x}^2}=\text{C}$
$\text{x}^2-\text{y}^2=\text{Cx}$

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 "Solve the Following Question.(5 Marks)" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If the derivative of $\tan^{-1} (a + bx)$ takes the value 1 at $x = 0$, prove that $1 + a^2 = b.$

Show that the vectors $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ given by $\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}},\ \vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ are non-coplanar. Express vector $\vec{\text{d}}=2\hat{\text{i}}-\hat{\text{j}}-3\hat{\text{k}}$ as a linear combination of the vectors $\vec{\text{a}},\ \vec{\text{b}}\text{ and }\vec{\text{c}}$.

Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}\frac{\sin3\text{x}}{\text{x}},&\text{if }\text{ x}\neq0\\4,&\text{if }\text{ x}=0\end{cases}$

Evaluate the following intregals:
$\int\frac{\text{ax}^2+\text{bx}+\text{c}}{(\text{x}-\text{a})(\text{x}-\text{b})(\text{x}-\text{c})}\ \text{dx},$ where a, b, c are distinct

Solve the following differential equation:
$(\text{x}+\text{y}+1)\frac{\text{dy}}{\text{dx}} = 1$

If $\text{u}=\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)$ and $\text{v}=\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big),$ where -1 < x < 1 then write the value of $\frac{\text{dy}}{\text{dx}}.$

Solve the following differential equation:
$\text{x}\frac{\text{dy}}{\text{dx}}+2\text{y}=\text{x}\cos\text{x}$

Using differentials, find the approximate values of the following:
$(33)^{\frac{1}{5}}$

Integrate the following functions w.r.t. x:

$x+\sqrt{5-4 \tau-r^2}$

The cartesian equation of a line are $3x + 1 = 6y - 2 =1 - z.$ Find the fixed point through which it passes, its direction ratios and also its vector equation.