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

Question

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

✓

Answer

$\text{x}^2\frac{\text{dy}}{\text{dx}}=\text{x}^2+\text{xy}+\text{y}^2$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{x}^2+\text{xy}+\text{y}^2}{\text{x}^2}$
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
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{x}^2+\text{x}^2\text{v}+\text{v}^2\text{x}^2}{\text{x}^2}$
$\Rightarrow\ \text{v + x}\frac{\text{dv}}{\text{dx}}=1+\text{v}+{\text{v}^2}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=\big(1+{\text{v}^2}\big)$
$\Rightarrow\ \frac{1}{1+\text{v}^2}\text{dv}=\frac{1}{\text{x}}\text{dx}$
Integrating both sides, we get
$\int\frac{1}{1+\text{v}^2}\text{dv}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\ \tan^{-1}{\text{v}}=\log|\text{x}|+\text{C}$
Putting $\text{v}=\frac{\text{y}}{\text{x}}$, we get
$\Rightarrow\ \tan^{-1}\Big(\frac{\text{y}}{\text{x}}\Big)=\log|\text{x}|+\text{C}$
Hence, $\tan^{-1}\Big(\frac{\text{y}}{\text{x}}\Big)=\log|\text{x}|+\text{C}$ 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

A box of constant volume $c$ is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?

Solve the following determinant equations:
$\begin{vmatrix}\text{x}+\text{a}&\text{b}&\text{c}\\\text{a}&\text{x}+\text{b}&\text{c}\\\text{a}&\text{b}&\text{x}+\text{c}\end{vmatrix}=0$

Show that the plane vector equation is $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}})=1$ and the line whose vector equation is $\vec{\text{r}}=(-\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})+\lambda(2\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}})$ are parallel. Also, find the distance between them.

Two sides of a triangle have lengths 'a' and 'b' and the angle between them is θ. What value of θ will maximize the area of the triangle? Find the maximum area of the triangle also.

Show that the following system of linear equation is inconsistent:
4x − 5y − 2z = 2
5x − 4y + 2z = −2
2x + 2y + 8z = −1

Differentiate the following functions with respect to x:
$\text{e}^{\text{x}\log\text{x}}$

Find the equatoion of the passing through the points (1, -1, 2) and (2, -2, 2) and which is perpendicular to the plane 6x - 2y + 2z = 9.

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{y}^2=4\text{ax}\text{ at }\Big(\frac{\text{a}}{\text{m}^2},\frac{2\text{a}}{\text{m}}\Big)$

A factory has three machines $\mathrm{X}, \mathrm{Y}$ and Z producing 1000,2000 and 3000 bolts per day respectively. The machine X produces $1 \%$ defective bolts, $Y$ produces $1.5 \%$ and $Z$ produces $2 \%$ defective bolts. At the end of a day, a bolt is drawn at random and is found to be defective. What is the probability that this defective bolt has been produced by machine X?

Evaluate the following integrals as limit of sum:$\int\limits^\text{b}_{\text{a}}\text{e}^{\text{x}}\text{ dx}$