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:
$\sin\Big(\frac{1+\text{x}^2}{1-\text{x}^2}\Big)$

If $\vec{\text{a}}=\text{a}_1\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}},\vec{\text{b}}=\text{b}_1\hat{\text{i}}+\text{b}_2\hat{\text{j}}+\text{b}_3\hat{\text{k}}$ and $\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}},$ then verify that $\vec{\text{a}}\times\big(\vec{\text{b}}+\vec{\text{c}}\big)=\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{a}}\times\vec{\text{c}}.$

Often it is taken that a truthful person commands, more respect in the society. $A$ man is known to speak the truth $4$ out of $5$ times. He throws a die and reports that it is a six. Find the probability that it is actually a six.Do you also agree that the value of truthfulness leads to more respect in the society?

Draw a rough sketch of the given curve y = 1 + |x + 1|, x = -3, x = 3, y = 0 and find the area of the region bounded by them, using integration.

Evaluate the following intregals:
$\int\frac{1}{\text{x}(\text{x}^\text{n}+1)}\text{ dx}$

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$\sin\text{ xy}+\cos(\text{x}+\text{y})=1$

Evaluate the following intregals:
$\int\frac{1}{\text{x}(\text{x}^3+8)}\ \text{dx}$

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

Evaluate the following: $\begin{vmatrix}\text{x}&1&1\\1&\text{x}&1\\1&1&\text{x}\end{vmatrix}$

Evaluate the integral in Exercise:
$\int\limits^{1}_{0}\sin^{-1}\bigg(\frac{2\text{x}}{1+\text{x}^{2}}\bigg)\text{dx}$