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

Question

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

✓

Answer

We have,
$\text{xy}\frac{\text{dy}}{\text{dx}}=\text{y}+2,\text{y}(2)=0$
$\Rightarrow\frac{\text{y}}{\text{y}+2}\text{dy}=\frac{1}{\text{x}}\text{dx}$
Integrating both sides, we get
$\int\frac{\text{y}}{\text{y}+2}\text{dy}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\int\frac{\text{y}+2-2}{\text{y}+2}\text{dy}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\int\text{dy}-2\int\frac{1}{\text{y}+2}\text{dy}=\log\text{x + C}$
$\Rightarrow\text{y}-2\log|\text{y}+2|=\log|\text{x}|+\text{C}\dots(1)$
It is given that at $\text{x}=2,\text{y}=0.$
Substituting the valuse of x and y in (1), we get
$-2\log2-\log2=\text{C}$
$\Rightarrow-\log(2^2\times2)=\text{C}$
$\Rightarrow\text{C}=-\log8$
Substituting the value of C in (1), we get
$\text{y}-2\log|\text{y}+2|=\log|\text{x}|-\log8$
$\Rightarrow\text{y}-2\log|\text{y}+2|=\log\Big|\frac{\text{x}}{8}\Big|$
Hence, $\text{y}-2\log|\text{y}+2|=\log\Big|\frac{\text{x}}{8}\Big|$ 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

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1\text{ at }(\text{a}\sec\theta,\text{b}\tan\theta)$

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

A manufacturer produces two types of steel trunks. He has two machines A and B. For completing, the first types of the trunk requires 3 hours on machine A and 3 hours on machine B, whereas the second type of the trunk requires 3 hours on machine A and 2 hours on machine B. Machines A and B can work at most for 18 hours and 15 hours per day respectively. He earns a profit of Rs. 30 and Rs. 25 per trunk of the first type and the second type respectively. How many trunks of each type must he make each day to make maximum profit?

Solve the following systems of homogeneous linear equations by matrix method:

3x + y - 2z = 0

Integrate the following w. r. t. x:

$\frac{12 x^2-2 x-9}{\left(4 x^2-1\right)(x+3)}$

Evaluate the following integrals:
$\int\frac{\text{e}^{\text{x}}(\text{x}-4)}{(\text{x}-2)^3}\text{dx}$

Solve the following differential equations:

$\left(x^2-y^2\right) d x+2 x y d y=0$

Prove the following results:
$2\sin^{-1}\frac{3}{5}=\tan^{-1}\frac{24}{7}$

Prove by vector method that the internal bisectors of the angles of a triangle are concurrent.

At what points on the following curves, is the tangent parallel to x-axis?
$\text{y}=\text{e}^{1-\text{x}^2}\text{ on }[-1,1]$