Solve the following differential equation dy dx -xe^x-5 2 + ^2x - Vidyadip

Question

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}-\text{x}\text{e}^\text{x}-\frac{5}{2}+\cos^2\text{x}$

✓

Answer

$\frac{\text{dy}}{\text{dx}}-\text{x}\text{e}^\text{x}-\frac{5}{2}+\cos^2\text{x}$
$\text{dy}=\Big(\text{xe}^\text{x}-\frac{5}{2}+\cos^2\text{x}\Big)\text{dx}$
$\int\text{dy}=\int\text{xe}^\text{x}\text{dx}-\frac{5}{2}\int\text{dx}+\cos^2\text{x dx}$
$\int\text{dy}=\int\text{xe}^\text{x}\text{dx}-\frac{5}{2}\int\text{dx}+\int\Big(\frac{1+\cos2\text{x}}{2}\Big)\text{dx}$
$=\int\text{xe}^\text{x}-\frac{5}{2}\int\text{dx}+\frac{1}{2}\int\text{dx}+\frac{1}{2}\int\cos2\text{x dx}$
$\int\text{dy}=\int\text{xe}^\text{x}-2\int\text{dx}+\frac{1}{2}\int\cos2\text{x dx}$
$\text{y}=[\text{x}\times\int\text{e}^\text{x}\text{dx}-\int(1\times\int\text{e}^\text{x}\text{dx})\text{dx}]-2\text{x}+\frac{1}{2}\frac{\sin2\text{x}}{2}+\text{C}$
Using integration by parts
$\text{y}=\text{xe}^\text{x}-\text{e}^\text{x}-2\text{x}+\frac{1}{4}\sin2\text{x}+\text{C}$
$\text{y}=\text{xe}^\text{x}-\text{e}^\text{x}-2\text{x}+\frac{1}{4}\sin2\text{x}+\text{C}$

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 "4 Marks" 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:
$\int\frac{\log\big(1+\frac{1}{\text{x}}\big)}{\text{x}(1+\text{x})}\text{dx}$

An automobile company uses three types of steel S₁, S₂ and S₃ for producing three types of cars C₁, C₂and C₃. Steel requirements (in tons) for each type of cars are given below:

Steel	Cars
	C₁	C₂	C₃
S₁	2	3	4
S₂	1	1	2
S₃	3	2	1

Using Cramer's rule, find the number of cars of each type which can be produced using 29, 13 and 16 tons of steel of three types respectively.

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{3}}_{-\frac{\pi}{3}}\frac{1}{1+\text{e}^{\tan\text{x}}}\text{ dx}$

If $\vec{\text{a}} \times \vec{\text{b}} = \vec{\text{c}} \times \vec{\text{d}}$ and $\vec{\text{a}} \times \vec{\text{c}} = \vec{\text{b}} \times \vec{\text{d}},$ show that $\vec{\text{a}} - \vec{\text{d}}$ is parallel to $\vec{\text{b}} - \vec{\text{c}},$ where $\vec{\text{a}} \neq \vec{\text{d}}$ and $\vec{\text{b}} \neq \vec{\text{c}}.$

If $\text{A}=\begin{bmatrix}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix},$ show that A^-1 = A³.

A square piece of tin of side 18cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{e}^{3\text{x}}\sin4\text{x}\times2^\text{x}$

Find the intervals in which the following functions are increasing or decreasing.
f(x) = 6 - 9x - x²

In order to supplement daily diet, a person wishes to take X and Y tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in X and Y are given as below:

Tablets	Iron	Calcium	Vitamin
X	6	3	2
Y	2	3	4

The person needs to supplement at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligrams of vitamins. The price of each tablet of X and Y is ₹ 2 and ₹1 respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically.

Find the equation of the plane which contains the line of intersection of the planes $\overrightarrow{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})-4=0,\overrightarrow{\text{r}}\cdot(\hat{\text{2i}}+\hat{\text{j}}-\hat{\text{k}})+5=0$and which is perpendicular to the plane$\overrightarrow{\text{r}}\cdot(5\hat{\text{i}}+3\hat{\text{j}}-6\hat{\text{k}})+8=0.$