Solve the following differential equation: y^2dx+(x^2-xy+y^2)dy=0 - Vidyadip

Question

Solve the following differential equation:
$\text{y}^2\text{dx}+(\text{x}^2-\text{xy}+\text{y}^2)\text{dy}=0$

✓

Answer

We have,
$\text{y}^2\text{dx}+(\text{x}^2-\text{xy}+\text{y}^2)\text{dy}=0$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{-\text{y}^2}{\text{x}^2-\text{xy}+\text{y}^2}$
This is a homogeneous differential equation.
Putting y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$, we get
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{-\text{v}^2\text{x}^2}{\text{x}^2-\text{vx}^2+\text{v}^2\text{x}^2}$
$\Rightarrow\ \text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{-\text{v}^2}{1-\text{v}+\text{v}^2}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=\frac{-\text{v}^2}{1-\text{v}+\text{v}^2}-\text{v}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=\frac{-\text{v}-\text{v}^3}{1-\text{v}+\text{v}^2}$
$\Rightarrow\ \frac{1-\text{v}+\text{v}^2}{\text{v}+\text{v}^3}\text{dv}=-\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\ \frac{1+\text{v}^2-\text{v}}{\text{v}(1+\text{v}^2)}\text{dv}=-\frac{1}{\text{x}}\text{dx}$
Integrating both sides, we get
$\int\frac{1+\text{v}^2-\text{v}}{\text{v}(1+\text{v}^2)}\text{dv}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\ \int\frac{1+\text{v}^2}{\text{v}(1+\text{v}^2)}\text{dv}-\int\frac{1}{\text{v}(1+\text{v}^2)}\text{dv}=-\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\ \int\frac{1}{\text{v}}\text{dv}-\int\frac{1}{1+\text{v}^2}\text{dv}=-\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\ \log|\text{v}|-\tan^{-1}|\text{v}|=-\log|\text{x}|+\log\text{C}$
$\Rightarrow\log\big|\frac{\text{vx}}{\text{C}}\big|=\tan^{-1}\text{v}$
$\Rightarrow\ \big|\frac{\text{vx}}{\text{C}}\big|=\text{e}^{\tan^{-1}\text{v}}$
Putting $\text{v}=\frac{\text{y}}{\text{x}}$, we get
$\Rightarrow\ |\text{y}|=\text{Ce}^{\tan^{-1}\text{v}}$
Hence, $|\text{y}|=\text{Ce}^{\tan^{-1}\text{v}}$ 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 "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

The radius of a sphere shrinks from 10 to 9.8cm. Find approximately the decrease in its volume.

Differentiate $\sin^{-1}\Big(4\text{x}\sqrt{1-4\text{x}^2}\Big)$ with respect to $\sqrt{1-4\text{x}^2},$ if:
$\text{x}\in\Big(-\frac{1}{2\sqrt{2}},\frac{1}{\sqrt{2\sqrt{2}}}\Big)$

An airline agrees to charter planes for a group. The group needs at least 160 first class seats and at least 300 tourist class seats. The airline must use at least two of its model 314 planes which have 20 first class and 30 tourist class seats. The airline will also use some of its model 535 planes which have 20 first class seats and 60 tourist class seats. Each flight of a model 314 plane costs the company Rs 100,000 and each flight of a model 535 plane costs Rs 150,000. How many of each type of plane should be used to minimize the flight cost? Formulate this as a LPP.

Find the coordinates of a point on the parabola $y=x^2+7 x+2$ which is closest to the strainght line $y=3 x-3$.

Show that the relation R on the set A = {x ∈ Z; 0 ≤ x ≤ 12}, given by R = {(a, b): a = b}, is an equivalence relation. Find the set of all elements related to 1.

A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.

Find the area of the region included between:

$y=x^2+3$ and $y=x+3$

$\text{A}=\begin{bmatrix}1&-2&0\\ 2&1&3\\ 0&-2&1\end{bmatrix}\text{and }\text{B}=\begin{bmatrix}7&2&-6\\ -2&1&-3\\ -4&2&5\end{bmatrix}$, find AB. Hence, solve the system of equations:
x - 2y = 10, 2x + y + 3z = 8 and -2y + z = 7

Differentiate the following functions with respect to x:
$\tan^{-1}\Big\{\frac{\text{x}}{\text{a}+\sqrt{\text{a}^2-\text{x}^2}}\Big\},-\text{a}<\text{x}<\text{a}$

Evaluate the following integrals:$\int^\limits1_0\sqrt{\frac{1-\text{x}}{1+\text{x}}}\text{ dx}$