Solve the following differential equation 1 x dy dx = ^ -1 x,x 0 - Vidyadip

Question

Solve the following differential equation
$\frac{1}{\text{x}}\frac{\text{dy}}{\text{dx}}=\tan^{-1}\text{x},\text{x}\neq0$

✓

Answer

$\frac{1}{\text{x}}\frac{\text{dy}}{\text{dx}}=\tan^{-1}\text{x},$$\text{dy}=\text{x}\tan^{-1}\text{x dx}$
$\int\text{dy}=\int\text{x}\tan^{-1}\text{x dx}$
$\text{y}-\tan^{-1}\text{x}\int\text{x dx}-\int\Big(\frac{1}{1+\text{x}^2}\int\text{x dx}\Big)\text{dx}+\text{C}$
Using intregration by part
$\text{y}=\frac{\text{x}^2}{2}\tan^{-1}\text{x}-\int\frac{\text{x}^2}{2(1+\text{x}^2)}\text{dx}+\text{C}$
$=\frac{\text{x}^2}{2}\tan^{-1}\text{x}-\frac{1}{2}\int\frac{\text{x}^2}{1+\text{x}^2}\text{dx}+\text{C}$
$=\frac{\text{x}^2}{2}\tan^{-1}\text{x}-\frac{1}{2}\int\Big(1-\frac{1}{\text{x}^2+1}\Big)\text{dx}+\text{C}$
$=\frac{\text{x}^2}{2}\tan^{-1}\text{x}-\frac{1}{2}\text{x}+\frac{1}{2}\tan^{-1}\text{x}+\text{C}$
$\text{y}=\frac{1}{2}(\text{x}^2+1)\tan^{-1}\text{x}-\frac{1}{2}\text{x}+\text{C}$

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

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{2^{\text{x}+1}}{1-4^{\text{x}}}\Big),-\infty<\text{x}<0$

Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 1 & 1 & 2 \\ 3 & 1 & 1 \\ 2 & 3 & 1 \end{bmatrix}$

Find the mean variance and standard deviation of the following probability distribution

$x_i$	$a$	$b$
$p_i$	$p$	$q$

Where $p + q = 1$

Amit's mathematics teacher has given him three very long lists of problems with the instruction to submit not more than 100 of them (correctly solved) for credit. The problem in the first set are worth 5 points each, those in the second set are worth 4 points each, and those in the third set are worth 6 points each. Amit knows from experience that he requires on the average 3 minutes to solve a 5 point problem, 2 minutes to solve a 4 point problem, and 4 minutes to solve a 6 point problem. Because he has other subjects to worry about, he can not afford to devote more than $3\frac{1}{2}$ hours altogether to his mathematics assignment. Moreover, the first two sets of problems involve numerical calculations and he knows that he cannot stand more than $2\frac{1}{2}$ hours work on this type of problem. Under these circumstances, how many problems in each of these categories shall he do in order to get maximum possible credit for his efforts? Formulate this as a LPP.

Let $\text{A} = \text{R} - \left\{3\right\}, \text{B} = \text{R} - \left\{1\right\}. \text{Let f : A} \rightarrow \text{B}$ be defined by $\text{f(x)}\frac{\text{x - 2}}{\text{x - 3}}, \forall \text{ x} \in \text{A}.$ Show that f is bijective. Also, find

$\text{x, if f}^{-1} \text{(x) = 4}$
$\text{f}^{-1} (7)$

Evaluate the following definite integral as limit of sum:
$\int_\limits{2}^{5} \ \text{x}^{2} \ \text{dx}$

If a, b and c are all non-zero and $\begin{vmatrix}1+\text{a}&1&1\\1&1+\text{b}&1\\1&1&1+\text{c} \end{vmatrix}=0,$ then prove that $\frac{1}{\text{a}}+\frac{1}{\text{b}}+\frac{1}{\text{c}}+1=0.$

By computing the shortest distance determine whether the following pairs of lines intersect or not:
$\frac{\text{x}-1}{2}=\frac{\text{y}+1}{3}=\text{z}$ and $\frac{\text{x}+1}{5}=\frac{\text{y}-2}{1};\text{z}=2$

In each of the show that the given differential equation is homogeneous and solve each of them. $(\text{x}-\text{y})\ \text{dy}- (\text{x}+\text{y})\ \text{dx}=0$

Find the vector equation of the line passing through the point (1, -1, 2) and perpendicular to the plane 2x - y + 3z - 5 = 0.