Solve the following differential equation (x-1) dy dx =2xy - Vidyadip

Question

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

✓

Answer

We have
$(\text{x}-1)\frac{\text{dy}}{\text{dx}}=2\text{xy}$
$\Rightarrow(\text{x}-1)\text{dy}=2\text{xy dx}$
$\Rightarrow\frac{2\text{x}}{(\text{x}-1)}\ \text{dx}=\frac{1}{\text{y}}\text{ dy}$
Integrating both sides, we get
$2\int\frac{\text{x}}{(\text{x}-1)}\ \text{dx}=\int\frac{1}{\text{y}}\ \text{dy}$
$\Rightarrow2\int\frac{\text{x}-1+1}{\text{x}-1}\ \text{dx}=\int\frac{1}{\text{y}}\ \text{dx}$
$\Rightarrow2\int\text{dx}+2\int\frac{1}{\text{x}-1}\text{dx}=\int\frac{1}{\text{y}}\ \text{dy}$
$\Rightarrow2\text{x}+2\log|\text{x}-1|=\log|\text{y}|+\text{C}$

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

Similar questions

Find the angle between the line joining the points (3, -4, -2) and (12, 2, 0) and the plane 3x - y + z = 1.

Find the maximum and the minimum values, if any, without using derivaives of the following functions:f(x) = -|x + 1| + 3 on R.

Let $\overrightarrow{\text{a}} = \hat{\text{i}} + 4\hat{\text{j}} +2\hat{\text{k}}, \overrightarrow{\text{b}} = 3\hat{\text{i}} - 2\hat{\text{j}} +7\hat{\text{k}}$ and $\overrightarrow{\text{c}} = 2\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}$ Find a vector $\overrightarrow{\text{d}}$ which is perpendicular to both $\overrightarrow{\text{a}} \text{and} \overrightarrow{\text{b}}\text{and} \overrightarrow{\text{c}} . \overrightarrow{\text{d}} = 27.$

Evaluate the following integrals:
$\int\frac{(\text{x}^2+1)(\text{x}^2+2)}{(\text{x}^2+3)(\text{x}^2+4)}\ \text{dx}$

If $\tan^{-1}\bigg(\frac{\text{x} - 2}{\text{x} - 4}\bigg) + \tan^{-1}\bigg(\frac{\text{x} + 2 }{\text{x} + 4 }\bigg) = \frac{\pi}{4} , $find the value of x.

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3}\text{cm}$ is $500\pi\text{cm}^{3}$ .

How should we choose two numbers, each greater than or equal to −2, whose sum so that the sum of the first and the cube of the second is minimum?

Evaluate the following integrals:
$\int_{0}^\limits{\pi}\frac{\sin\text{x}}{\sin\text{x}+\cos\text{x}}\text{ dx}$

Solve the following systems of linear equations by cramer's rule:

Show that the semi $-$ vertical angle of a cone of maximum volume and given slant height is $\tan ^{-1} \sqrt{2}$ or $\cos ^{-1} \frac{1}{\sqrt{3}}$.