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

Question

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

✓

Answer

We have,
$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{y}^2$
$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}^2-\text{y}$
$\Rightarrow\frac{1}{\text{y}^2-\text{y}}\ \text{dy}=\frac{1}{\text{x}}\ \text{dx}$
integrating both sides, we get
$\int\frac{1}{\text{y}^2-\text{y}}\text{dy}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\int\frac{1}{\text{y}(\text{y}-1)}=\text{dy}=\int\frac{1}{\text{x}}\text{dx}\ ...(1)$
Let $\frac{1}{(\text{y}-1)}=\frac{\text{A}}{\text{y}}+\frac{\text{B}}{\text{y}-1}$
$\Rightarrow1=\text{A}(\text{y}-1)+\text{B}(\text{y})$
putting y = 0, we get
1 = -A
⇒ A = -1
putting y = 1, we get
1 = B
$\therefore\frac{1}{\text{y}(\text{y}-1)}=\frac{-1}{\text{y}}+\frac{1}{\text{y}-1}$
$\Rightarrow\int\frac{1}{\text{y}(\text{y}-1)}\text{dy}=\int\frac{-1}{\text{y}}\text{dy}+\int\frac{1}{\text{y}-1}\text{dy}\ ...(2)$
From (1) & (2) , we get
$\int\frac{-1}{\text{y}}\text{dy}+\int\frac{1}{\text{y}-1}\text{dy}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow-\log|\text{y}|+\log|\text{y}-1|=\log|\text{x}|+\log\text{C}$
$\Rightarrow\log\Big|\frac{\text{y}-1}{\text{y}}\Big|-\log|\text{x}|=\log\text{C}$
$\Rightarrow\log\Big|\frac{\text{y}-1}{\text{xy}}\Big|=\log\text{C}$
$\Rightarrow\frac{\text{y}-1}{\text{xy}}=\text{C}$
$\Rightarrow\text{y}-1=\text{Cxy}$
hence, $\text{y}-1=\text{Cxy}$ 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 "4 Marks" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

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

Solve the following differential equation
$\text{xy}(\text{y}+1)\text{dy}=(\text{x}^2+1)\text{dx}$

Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, -3), B(-2, -3, 5) nad C(5, 3, -3).

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=(\tan\text{x})^{\log\text{x}}+\cos^2\big(\frac{\pi}{4}\big)$

An automobile company uses three types of steel S₁, S₂ and S₃ for producing three types of cars C₁, C₂and C₃. Steel requirements (in tons) for each type of cars are given below:

Steel	Cars
	C₁	C₂	C₃
S₁	2	3	4
S₂	1	1	2
S₃	3	2	1

Using Cramer's rule, find the number of cars of each type which can be produced using 29, 13 and 16 tons of steel of three types respectively.

From a lot of 30 bulbs that includes 6 defective bulbs, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

If $\text{A}=\begin{bmatrix}1&-1&0\\ 2&3&4\\ 0&1&2\end{bmatrix}\text{and }\text{B}=\begin{bmatrix}2&2&-4\\ -4&2&-4\\ 2&-1&5\end{bmatrix}$ are two square matrices, find AB and hence solve the system of linear equations:
x - y = 3, 2x + 3y + 4z = 17, y + 2z = 7

Prove that x² - y² = c(x² + y²)² is the general solution of differential equation $(x^3-3xy^2)dx=(y^3-3x^2y)dy$, where c is a parameter.

Find the particular solution, satisfying the given condition, for the following differential equation:

$\frac{\text{dy}}{\text{dx}} - \frac{\text{y}}{\text{x}} + \text{cosec} \bigg(\frac{\text{y}}{\text{x}}\bigg) = \text {0; y = 0 when x} = 1.$

Solve the following differential equation
$\sin^4\text{x}\frac{\text{dy}}{\text{dx}}=\cos\text{x}$