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

Question

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

✓

Answer

$\text{2x}^{2}\frac{\text{dy}}{\text{dx}}-\text{2 xy + y}^{2}=0\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{2 xy - y}^{2}}{\text{2x}^{2}}=\frac{\text{2}\frac{\text{y}}{\text{x}}-\frac{\text{y}^{2}}{\text{x}^{2}}}{\text{2}}$Putting $\frac{\text{y}}{\text{x}}$ v so that y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v} + \text{x}\ \frac{\text{dv}}{\text{dx}}$
$\therefore\text{v + x }\frac{\text{dv}}{\text{dx}}=\text{v}-\frac{1}{2}\text{v}^{2}\therefore \text{x}\frac{\text{dv}}{\text{dx}}=-\frac{1}{2}\text{v}^{2}$
$\Rightarrow 2\int\frac{\text{dv}}{\text{v}^{2}} = -\int\frac{\text{dx}}{\text{x}}\Rightarrow\frac{2}{\text{v}}=\log\text{ x + c}$
$\therefore 2 \frac{\text{x}}{\text{y}}=\log\text{ x + c or y}=\frac{\text{2x}}{\text{log x + c}}.$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

A class has 15 students whose ages are 14, 17, 15, 14, 21, 19, 20, 16, 18, 17, 20, 17, 16, 19 and 20 years respectively. One student is selected in such a manner that each has the same chance to being selected and the age X of the selected student is recorded. What is the probability distribution of the random variable X.

Find matrix X so that $\text{X} \begin{pmatrix} 1 & 2 & 3 \\ \\ 4 & 5 & 6 \end{pmatrix} = \begin{pmatrix} -7 & -8 & -9 \\ \\ 4 & 5 & 6 \end{pmatrix}.$

Show that the function g(x) = x - [x] is discontinuous at all integral points. Here [x] denotes the greatest integer function.

Evaluate the following integrals:
$\int^\limits{\text{a}}_0\text{x}\sqrt{\frac{\text{a}^2-\text{x}^2}{\text{a}^2+\text{x}^2}}\text{ dx}$

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

Find the equation of the normal lines to the curve $3x^2 - y^2 = 8$ which are parallel to the line $x + 3y = 4.$

Using properties of determinants, show that $\triangle\text{ABC}$ is isosceles if:
$\begin{vmatrix} 1 & 1 & 1 \\ 1 + \cos\text{A} & 1 + \cos\text{B} & 1 + \cos\text{C} \\ \cos^{2}\text{A} + \cos\text{A} & \cos^{2}\text{B}+\cos\text{B} & \cos^{2}\text{C} + \cos\text{C} \end{vmatrix} = 0 $

Find the angle between line $\frac{\text{x}-1}{1}=\frac{\text{y}-2}{-1}=\frac{\text{z}+1}{1}$ and the plane $2x + y - z = 4.$

If $\text{y}=\text{x}^\text{n}\{\text{a}\cos(\log\text{x})+\text{b}\sin(\log\text{x})\},$ prove that $\text{x}^2\frac{\text{d}^2\text{y}}{\text{dx}^2}+(1-2\text{n})\frac{\text{dy}}{\text{dx}}+(1+\text{n}^2)\text{y}=0.$

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes.