Solve the following differential equations: 2(y+3)-xy dy dx =0,y(… - Vidyadip

Question

Solve the following differential equations:
$2(\text{y}+3)-\text{xy}\frac{\text{dy}}{\text{dx}}=0,\text{y}(1)=-2$

✓

Answer

$2(\text{y}+3)-\text{xy}\frac{\text{dy}}{\text{dx}}=0$
$\Rightarrow2(\text{y}+3)=\text{xy}\frac{\text{dy}}{\text{dx}}$
$\Rightarrow\frac{2}{\text{x}}\text{dx}=\frac{\text{y}}{\text{y}+3}\text{dy}$
$\Rightarrow\frac{2}{\text{x}}\text{dx}=\frac{\text{y}+3-3}{\text{y}+3}\text{dy}$
$\Rightarrow\frac{2}{\text{x}}\text{dx}=\Big(1-\frac{3}{\text{y}+3}\Big)\text{dy}$
$\Rightarrow\int\frac{2}{\text{x}}\text{dx}=\int\Big(1-\frac{3}{\text{y}+3}\Big)\text{dy}$
$\Rightarrow2\log\text{x = y}-3\log|\text{y}+3|+\text{C}$
$\Rightarrow\log\text{x}^2+\log|(\text{y}+3)^3|=\text{y + C}$
$\Rightarrow\log|(\text{x}^2)(\text{y}+3)^3|=\text{y + C}...(1)$
$\Rightarrow\log|(1)^2(-2+3)^3|=-2+\text{C}$
$\Rightarrow\text{C}=2$
Substituting the value of C in (1), we get
$\log|(\text{x}^2)(\text{y}+3)^3|=\text{y}+2$
$\Rightarrow(\text{x}^2)(\text{y}+3)^3=\text{e}^{\text{y}+2}$

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 systems of homogeneous linear equations by matrix method:

3x + 6y - 5z = 0

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\frac{\text{y}}{\text{x}}=\text{x}^3$

Differentiate the following functions with respect to x:
$\text{e}^{3\text{x}}\cos2\text{x}$

Evaluate the following intregals:
$\int\frac{5\text{x}}{(\text{x}+1)(\text{x}^2-4)}\text{dx}$

If A is a square matrix, using mathematical induction prove that (A^T)ⁿ = (Aⁿ)^T for all n ∈ N.

In a dice game, a player pays a stake of Rs. 1 for each throw of a die. She receives Rs. 5 if the die shows a 3, Rs. 2 if the die shows a 1 or 6, and nothing otherwise. What is the player’s expected profit per throw over a long series of throws?

A company sells two different products, A and B. The two products are produced in a common production process, which has a total capacity of 500 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of unit of A that can be sold is 70 and that for B is 125. If the profit is Rs. 20 per unit for the product A and Rs. 15 per unit for the product B, how many units of each product should be sold to maximize profit?

Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector $2\hat{i} + 3\hat{j} + 4\hat{k}$ to the plane $\overrightarrow{\text{r}}. (2\hat{i} + \hat{j} + 3\hat{k}) - 26 = 0.$ Also find image of P in the plane.

Solve the following initial value problems:
$(\text{y}^4-2\text{x}^3\text{y})\text{dx}+(\text{x}^4-2\text{xy}^3)\text{dy}=0,\text{y}(1)=1$

Show that $\text{y}=\text{ae}^{2\text{x}}+\text{be}^{-\text{x}}$ is a solution of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-\frac{\text{dy}}{\text{dx}}-2\text{y}=0$