Solve the following differential equation: dy dx =y -2 - Vidyadip

Question

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\text{y}\tan\text{x}-2\sin\text{x}$

✓

Answer

Here,$\frac{\text{dy}}{\text{dx}}=\text{y}\tan\text{x}-2\sin\text{x}$
It is a linear differential equation. Comparing the equation by,
$\text{P}=-\tan\text{x},\text{Q}=-\sin\text{x}$
I.F. $=\text{e}^{\int\text{Pdx}}$
$=\text{e}^{-\int\tan\text{xdx}}$
$=\text{e}^{-\log\sec\text{x}}$
$=\frac{1}{\sec\text{x}}$
Solution of the equation is given by,
$\text{y}\times(\text{I.F.})=\int\text{Q}\times(\text{I.F.})\text{dx + C}$
$\frac{\text{y}}{\sec\text{x}}=\int-\frac{2\sin\text{x}}{\sec\text{x}}\text{dx+ C}$
$\text{y}\cos\text{x}=-\int2\sin\text{x}\cos\text{xdx + C}$
$\text{y}\cos\text{x}=-\int\sin2\text{xdx + C}$
$\text{y}\cos\text{x}=\frac{\cos2\text{x}}{2}+\text{C}$
$\text{y}=\frac{\cos2\text{x}}{2\cos\text{x}}+\frac{\text{C}}{\cos\text{x}}$

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

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{5\text{x}}{1+6\text{x}^3}\Big), -\frac{1}{\sqrt{6}}<\text{x}<\frac{1}{\sqrt{6}}$

Extend the definition of the following by continuity $\text{f(x)}=\frac{1-\cos7(\text{x}-\pi)}{5(\text{x}-\pi)^2}$ at the point $\text{x}=\pi.$

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

Two sides of a triangle have lengths 'a' and 'b' and the angle between them is θ. What value of θ will maximize the area of the triangle? Find the maximum area of the triangle also.

If $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ 2\hat{\text{i}}+5\hat{\text{J}},\ 3\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}$ and $\hat{\text{i}}-6\hat{\text{j}}-\hat{\text{k}}$ respectively are the position vectors of points $A, B, C$ and $D,$ then find the angle between the straight lines $AB$ and $CD.$ Find whether $\vec{\text{AB}}$ and $\vec{\text{CD}}$ are collinear or not.

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

A and B throw a pair of dice alternately. A wins the game if he gets a total of 6 and B wins if she gets a total of 7. It A starts the game, find the probability of winning the game by A in third throw of the pair of dice.

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

Solve the following initial value problems:
$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{x}\cos\text{x}+\sin\text{x},\text{ y}\Big(\frac{\pi}{2}\Big)=1$

Find the coordinates of the foot of the perpendicular and the perpendicular distance of the point P(3, 2, 1) from the plane 2x - y + z + 1 = 0. Also, find the image of the point in the plane.