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

Question

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}=\sin\text{x}\cos\text{x}$

✓

Answer

We have,
$\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}=\sin\text{x}\cos\text{x}\ \dots(1)$
Clearly, it is a linear differential equation of the form
$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
where
$\text{P}=\cos\text{x}$
$\text{Q}=\sin\text{x}\cos\text{x}$
$\therefore$ I.F. $=\text{e}^{\int\text{Pdx}}$
$=\text{e}^{\int\cos\text{xdx}}$
$=\text{e}^{\sin\text{x}}$
Multiplying both sides of (1) by $\text{e}^{\sin\text{x}}$ we get
$\text{e}^{\sin\text{x}}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}\Big)=\text{e}^{\sin\text{x}}\sin\text{x}\cos\text{x}$
$\Rightarrow\ \text{e}^{\sin\text{x}}\frac{\text{dy}}{\text{dx}}+\text{e}^{\sin\text{x}}\text{y}\cos\text{x}=\text{e}^{\sin\text{x}}\sin\text{x}\cos\text{x}$
Integrating both sides with respect to x, we get
$\text{y}\text{e}^{\sin\text{x}}=\int\text{e}^{\sin\text{x}}\sin\text{x}\cos\text{x dx + C}$
$\Rightarrow\ \text{y}\text{e}^{\sin\text{x}}=\text{I + C}\ \dots(2)$
where
$\text{I}=\int\text{e}^{\sin\text{x}}\sin\text{x}\cos\text{x dx}$
Putting $\text{t}=\sin\text{x},$ we get
$\text{dt}=\cos\text{x dx}$
$\therefore\ \text{I}=\int\text{e}^{\text{t}}\text{t dt}$
$=\text{t}\int\text{e}^{\text{t}}\text{dt}-\int\Big[\frac{\text{d}}{\text{dt}}(\text{t})\int\text{e}^{\text{t}}\text{dt}\Big]\text{dt}$
$=\text{te}^{\text{t}}-\text{e}^{\text{t}}$
$=\text{e}^{\text{t}}(\text{t}-1)$
$=\text{e}^{\sin\text{x}}(\sin\text{x}-1)$
Putting the value of I in (2), we get
$\text{ye}^{\sin\text{x}}=\text{e}^{\sin\text{x}}(\sin\text{x}-1)+\text{C}$
$\Rightarrow\ \text{y}=\sin\text{x}-1+\text{Ce}^{-\sin\text{x}}$
Hence, $\text{y}=\sin\text{x}-1+\text{Ce}^{-\sin\text{x}}$ is the required solution.

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

Integrate the rational function $\frac{1}{{{x^4} - 1}}$

Find the area enclosed by the curve $y = -x^2$ and the strainght line $x + y + 2 = 0.$

Find the equation of the plane passing through (a, b, c) and parallel to the plane $\vec{\text{r}}\cdot(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})=2.$

If the vectors $\vec{\text{a}}=2\hat{\text{i}}-3\hat{\text{j}}$ and $\vec{\text{b}}=-6\hat{\text{i}}+\text{m}\hat{\text{j}}$ are collinear, find tghe value of m.

An automobile manufacturer makes automobiles and trucks in a factory that is divided into two shops. Shop A, which performs the basic assembly operation, must work 5 man-days on each truck but only 2 man-days on each automobile. Shop B, which performs finishing operations, must work 3 man-days for each automobile or truck that it produces. Because of men and machine limitations, shop A has 180 man-days per week available while shop B has 135 man-days per week. If the manufacturer makes a profit of Rs 30000 on each truck and Rs 2000 on each automobile, how many of each should he produce to maximize his profit? Formulate this as a LPP.

$A$ purse contains $2$ silver and $4$ copper coins. $A$ second purse contains $4$ silver and $3$ copper coins. If a coin is pulled at random from one of the two purses, what is the probability that it is a silver coin?

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

Find an equation for the set all points that are equidistant from the planes $3x - 4y + 12z = 6$ and $4x + 3z = 7$

Find the matrix $X$ satisfying the matrix equation. $\text{X}\begin{bmatrix}5 & 3 \\-1 & -2 \end{bmatrix}\begin{bmatrix}14 & 7 \\7 & 7 \end{bmatrix}$

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

5x - 7y + z = 11,

6x - 8y - z = 15,

3x + 2y - 6z = 7