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

Question

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

✓

Answer

We have,
$\frac{\text{dy}}{\text{dx}}+\text{y}\tan\text{x}=\cos\text{x}\ \dots(1)$
Clearly, it is a linear differential equation of the form
$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
where
$\text{P}=\tan\text{x}$
$\text{Q}=\cos\text{x}$
$\therefore$ I.F. $=\text{e}^{\int\text{Pdx}}$
$=\text{e}^{\int\tan\text{xdx}}$
$=\text{e}^{\log|\sec\text{x}|}=\sec\text{x}$
Multiplying both sides of (1) by $\sec\text{x},$ we get
$\sec\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}\tan\text{x}\Big)=\cos\text{x}\times\sec\text{x}$
$\Rightarrow\ \sec\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\sec\text{x}\tan\text{x}=1$
Integrating both sides with respect to x, we get
$\text{y}\sec\text{x}=\int\text{dx + C}$
$\Rightarrow\ \text{y}\sec\text{x}=\text{x + C}$
Hence, $\text{y}\sec\text{x}=\text{x + C}$ 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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Show that the points A(-1, 4, -3), B(3, 2, -5), C(-3, 8, -5) and D(-3, 2, 1) are coplanar.

Prove that the least perimeter of and isosceles triangle in which a circle of radius r can be inscribed is $6\sqrt{3}\text{ r}.$

Evaluate the following intregals:
$\int\frac{2}{2+\sin^22\text{x}}\text{ dx}$

Find the length and the foot of perpendicular from the poin $\Big(1,\frac{3}{2},2\Big)$ to the plane 2x - 2y + 4z + 5 = 0.

Find the equation of the tangent to the curve $x^2 + 3y − 3 = 0,$ which is parallel to the line $y = 4x − 5.$

Of all the closed cylindrical cans (right circular), which enclose a given volume of $100cm^3,$ which has the minimum surface area ?

Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 2x^3 - 24x + 107$

The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.

The probability distribution of a random variable X is given below:

$\text{X}$	$0$	$1$	$2$	$3$
$\text{P}(\text{X})$	$\text{k}$	$\frac{\text{k}}{2}$	$\frac{\text{k}}{4}$	$\frac{\text{k}}{8}$

Determine the value of k.
Determine $\text{P}(\text{X}\leq2)$ and $\text{P}(\text{X}\geq2)$
Find $\text{P}(\text{X}\leq2)+\text{P}(\text{X}\geq2)$

A fair die is tossed. Let X denote 1 or 3 according as an odd or an even number appears. Find the probability distribution, mean and variance of X.