Download App

Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\text{y}=\cos\text{x}$
✓

Answer

Here, $\frac{\text{dy}}{\text{dx}}+\text{y}=\cos\text{x}$
It is a linear differential equation. Comparing the equation by,
$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
$\text{P}=1,\text{Q}=\cos\text{x}$
I.F. $=\text{e}^{\int\text{Pdx}}$
$=\text{e}^{\int\text{dx}}$
$=\text{e}^{\text{x}}$
Solution of the equation is given by,
$\text{y}\times(\text{I.F.})=\int\text{Q}\times(\text{I.F.})\text{dx + C}$
$\text{y}(\text{e}^{\text{x}})=\int(\cos\text{x})(\text{e}^{\text{x}})+\text{C}_1\ \dots(\text{i})$
Let $\text{I}=\int\text{e}^{\text{x}}\cos\text{xdx}$
$=\cos\text{x}\times\int\text{e}^{\text{x}}\text{dx}\int(\sin\text{x}\int\text{e}^{\text{x}}\text{dx})\text{dx + C}_2$
Using integration by parts
$\text{I}=\text{e}^{\text{x}}\cos{\text{x}}+\int\sin\text{xe}^{\text{x}}\text{dx + C}$
$=\text{e}^{\text{x}}\cos\text{x}+\big[\sin\text{x}\int\text{e}^{\text{x}}\text{dx}-\int(\cos\text{x}\int\text{e}^{\text{x}}\text{dx})\text{dx}\big]+\text{C}_2$
$\text{I}=\text{e}^{\text{x}}\cos\text{x}+\sin\text{e}^\text{x}-\text{I}+\text{C}_2$
$2\text{I}=\text{e}^{\text{x}}(\cos\text{x}+\sin\text{x})+\text{C}_2$
$\text{I}=\frac{\text{e}^{\text{x}}}{2}(\cos\text{x}+\sin\text{x})+\frac{\text{C}_2}{2}$
$\text{I}=\frac{\text{e}^{\text{x}}}{2}(\cos\text{x}+\sin\text{x})+\text{C}_3$
Putting I in equation (i),
$\text{ye}^{\text{x}}=\frac{\text{e}^{\text{x}}}{2}(\cos\text{x}+\sin{\text{x}})+\text{C}_1+\text{C}_3$
$\text{ye}^{\text{x}}=\frac{\text{e}^{\text{x}}}2(\cos\text{x}+\sin\text{x})+\text{C}$
$\text{y}=\frac{1}2(\cos\text{x}+\sin\text{x})+\text{Ce}^{-\text{x}}$

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\tan^{-1}\sqrt{\frac{1-\text{x}}{1+\text{x}}}\text{dx}$
→
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{x}^{\sin\text{x}}+\big(\sin\text{x}\big)^\text{x}$
→
A small firm manufacturers items A and B. The total number of items A and B that it can manufacture in a day is at the most 24. Item A takes one hour to make while item B takes only half an hour. The maximum time available per day is 16 hours. If the profit on one unit of item A be Rs. 300 and one unit of item B be Rs. 160, how many of each type of item be produced to maximize the profit? Solve the problem graphically.
→
Find perpendicular distance of the point $(1, 0, 0)$ from the line $\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8}$ .Also, find the coordinates of the foot of the perpendicular and the equation of the perpendicular.
→
If the co-ordinates of the vertices of an equilateral triangle with sides of length ‘a’ are $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3)$, then $\begin{vmatrix}\text{x}_1&\text{y}_1&1\\\text{x}_1&\text{y}_2&1\\\text{x}_2 &\text{y}_3&1\end{vmatrix}^2=\frac{3\text{a}^4}{4}.$
→
Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$y^3 - 3xy^2 = x^3 + 3x^2y$
→
Evaluate the following integrals:$\int\frac{\text{x}^2}{\text{x}^2+6\text{x}+12}\text{ dx}$
→
If $\text{x}=\text{a}(\cos\text{t}+\text{t}\sin\text{t})\ \text{and}\ \text{y}=\text{a}(\sin\text{t}-\text{t}\cos\text{t}),$ find the value of $\frac{\text{d}^2\text{y}}{\text{dx}^2}\ \text{at}\ \text{t}=\frac{\pi}{4}.$
→
Show that each of the given three vectors is a unit vector:$\frac{1}{7}(2\hat{i}+3\hat{j}+6\hat{k}),\ \frac{1}{7}(3\hat{i}-6\hat{j}+2\hat{k}),\ \frac{1}{7}(6\hat{ i}+2\hat{j}-3\hat{k})$
Also, show that they are mutually perpendicular to each other.
→
Find the point on the curve $y = x^2 - 2x + 3,$ where the tangent is parallel to x-axis.
→