Solve the following differential equation : ( ^ –1 y + x) dy = (1… - Vidyadip

Question

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

✓

Answer

$\frac{\text{dx}}{\text{dy}}-\frac{\text{x}}{1+\text{y}^2}=\frac{\cot^{-1}}{1+\text{y}^{2}}$
$\text{I.F.}=\text{e}^{-\int\frac{\text{x}}{1+\text{y}^2}}=\text{e}^{\cot^{-1}\text{y}}$
$\text{x}.\text{e}^{\cot^{-1}\text{y}}=\int\frac{\cot^{-1}\text{y}\ \text{e}^{\cot^{-1}\text{y}}}{1+\text{y}^2}\text{dy}$
Integrating, we get
$\text{x}.\text{e}^{\cot^{-1}\text{y}}=\int\frac{\cot^{-1}\text{y}\ \text{e}^{\cot^{-1}\text{y}}}{1+\text{y}^2}\text{dy}$
put $\cot^{–1} y = t$
$=-\int\text{t }\text{e}^{\text{t}}\text{dt}$
$= (1 – t) e^t + c$
$\Rightarrow x = (1 – \cot^{–1}y) + ce^{–\cot–1 y}$

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

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{(\text{x}-\text{y})+3}{2(\text{x}-\text{y})+5}$

Classify the following functions as injection, surjection or bijection: $f : R \rightarrow R,$ defined by $\text{f(x)}=\frac{{x}}{{x}^2+1}$

Differentiate the following functions with respect to x:
$\frac{\text{e}^{2\text{x}}+\text{e}^{-2\text{x}}}{\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}}$

Find the probability of 4 turning up at least once in two tosses of a fair die.

If X follows a binomial distribution with mean 4 and variance 2, find P (X ≥ 5).

Find the shortest distance between the lines
$\vec{\text{r}}=\big(\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$ and, $\vec{\text{r}}=2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\mu\big(2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}\big)$

Evaluate the following integrals:
$\int\text{cosec}^43\text{x}\text{ dx}$

In a culture, the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present?

Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=\big(\hat{\text{i}}+\hat{\text{j}}\big)+\lambda\big(2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$ and $\vec{\text{r}}=2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}+\mu\big(3\hat{\text{i}}-5\hat{\text{j}}+2\hat{\text{k}}\big)$

The slope of the tangent at a point P(x, y) on a curve is $\frac{-\text{x}}{\text{y}}$. If the curve passs es through the point (3, -4). Find the equation of the curve.