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

Question

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}-\sqrt{\frac{\text{y}^2}{\text{x}^2}-1}$

✓

Answer

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

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

For the following pairs of matrices verify that $(AB)^{-1} = B^{-1} A^{-1}:$
$\text{A}=\begin{bmatrix}3 & 2 \\7 & 5 \end{bmatrix}\text{ and B}=\begin{bmatrix}4 & 6 \\3 & 2 \end{bmatrix}$

Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P wants to award ₹ x each, ₹ y each and ₹ z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹ 2,200. School Q wants to spend ₹ 3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each values is ₹ 1,200, using matrices, find the award money for each value.
Apart from these three values, suggest one more value which should be considered for award.

Using integration find the area of the triangular region whose sides have equations y = 2x + 1, y = 3x + 1 and x = 4.

In the following, determine the values of constants involved in the definition so that the given function is continuous:
$\text{f(x)}=\begin{cases}\frac{\sqrt{1+\text{px}}\sqrt{1-\text{px}}}{\text{x}},&\text{if }-1\leq\text{ x}\leq-0\\\frac{2\text{x}+1}{\text{x}-2},&\text{if }0\leq\text{ x}\leq1\end{cases}$

Differentiate the following functions from first principles:
$\log\text{cosec x}$

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\text{e}^{\text{x}}\cos{\text{x}}\text{ on }\Big[\frac{-\pi}{2},\frac{\pi}{2}\Big]$

Find the equation of the curve that passes through the point (0, a) and is such that at any point (x, y) on it, the product of its slope and the ordinate is equal to the abscissa.

Without expanding, show that the values of the following determinant are zero: $\begin{vmatrix}6&-3&2\\2&-1&2\\-10&5&2 \end{vmatrix}$

An open box with square base is to be made of a given quantity of card board of area $c^2.$ Show that the maximum volume of the box is $\frac{\text{c}^2}{6\sqrt{3}}$ cubic units.

$\text{If y}=3\cos(\log \text{x})+4\sin(\log\text{x}),\text{ show that }\text{x}^2\text{y}_2+\text{xy}_1+\text{y}=0$