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

Question

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

✓

Answer

Here, $(1+\text{x}^2)\frac{\text{dy}}{\text{dx}}-2\text{xy}=(\text{x}^2+2)(\text{x}^2+1)$ $\frac{\text{dy}}{\text{dx}}-\frac{2\text{x}}{\text{x}^2+1}\text{y}=(\text{x}^2+2)$ It is a linear differential equation. Comparing it with, $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$ $\text{P}=-\frac{2\text{x}}{\text{x}^2+1},\text{Q}=\text{x}^2+2$ I.F. $=\text{e}^{\int\text{Pdx}}$ $=\text{e}^{-\int\frac{2\text{x}}{\text{x}^2+1}\text{dx}}$ $=\text{e}^{-\log|\text{x}^2+1|}$ $=\frac{1}{(\text{x}^2+1)}$Solution of the equation is given by,
$\text{y}\times(\text{I.F.})=\int\text{Q}\times(\text{I.F.})\text{dx + C}$ $\text{y}\Big(\frac{1}{\text{x}^2+1}\Big)=\int\Big(\frac{\text{x}^2+2}{\text{x}^2+1}\Big)\text{dx + C}$ $=\int\Big(1+\frac{1}{\text{x}^2+1}\Big)\text{dx}+\text{C}$ $\frac{\text{y}}{(\text{x}^2+1)}=\text{x}+\tan^{-1}\text{x + C}$ $\text{y}=(\text{x}^2+1)(\text{x}+\tan^{-1}\text{x + C})$

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 "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 $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{a}(\cos\theta+\theta\sin\theta)$ and $\text{y}=\text{a}(\sin\theta-\theta\sin\theta-\theta\cos\theta)$

Maximize Z = 3x + 3y, if possible,
Subject to the constraints
$\text{x}-\text{y}\leq1$
$\text{x}+\text{y}\geq3$
$\text{x},\text{y}\geq0$

Find the cartesian equation of a line passing through (1, -1, 2) and parallel to the line whose equation are $\frac{\text{x}-3}{1}=\frac{\text{y}-1}{2}=\frac{\text{z}+1}{-2}.$ Also, reduce the equation obtained in vector form.

In a multiple-choice examination with three possible answers for each of the five questions out of which only one is correct, what is the probability that a candidate would get four or more correct answers just by guessing?

Solve the following system of homogeneous linear equations:

3x + y + z = 0,

x - 4y + 3z = 0,

2x + 5y - 2z = 0

Find the area bounded by the parabola $y^2 = 4x$ and the line $y = 2x - 4:$
By using horizontal strips.

A wire of length 28m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the circle and the square is minimum?

Evaluate the following integrals:
$\int\text{x}\sqrt{\text{x}^4+1}\text{dx}$

Evaluate the following intregals:
$\int\frac{\text{x}^3}{(\text{x}-1)(\text{x}-2)(\text{x}-3)}\ \text{dx}$

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\text{e}^{1-\text{x}^2}\text{ on }[-1,1]$