Solve the following differential equation: dy dx +2y=6e^ x - Vidyadip

Question

Solve the following differential equation:$\frac{\text{dy}}{\text{dx}}+2\text{y}=6\text{e}^{\text{x}}$

✓

Answer

We have,
$\frac{\text{dy}}{\text{dx}}+2\text{y}=6\text{e}^{\text{x}}\ \dots(1)$
Clearly, it is a linear differential equation of the form
$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
where
$\text{P}=2$
$\text{Q}=6\text{e}^{\text{x}}$
$\therefore$ I.F. $=\text{e}^{\int\text{Pdx}}$
$=\text{e}^{\int2\text{dx}}$
$=\text{e}^{2\text{x}}$
Multiplying both sides of (1) by $\text{e}^{2\text{x}},$ we get
$\text{e}^{2\text{x}}\Big(\frac{\text{dy}}{\text{dx}}+2\text{y}\Big)=6\text{e}^{2\text{x}}\text{e}^\text{x}$
$\Rightarrow\ \text{e}^{2\text{x}}\frac{\text{dy}}{\text{dx}}+2\text{e}^{2\text{x}}\text{y}=6\text{e}^{3\text{x}}$
Integrating both sides with respect to x, we get
$\text{y}\text{e}^{2\text{x}}=6\int\text{e}^{3\text{x}}\text{dx + C}$
$\Rightarrow\ \text{y}\text{e}^{2\text{x}}=6\frac{\text{e}^{3\text{x}}}3+\text{C}$
$\Rightarrow\ \text{ye}^{2\text{x}}=2\text{e}^{3\text{x}}+\text{C}$
Hence, $\text{ye}^{2\text{x}}=2\text{e}^{3\text{x}}+\text{C}$ 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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

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

Evaluate the following integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\frac{1}{5+4\sin\text{x}}\text{ dx}$

By using the properties of definite integrals, evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x d x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}$

Find the angle between the lines whose direction cosines are given by the equations $l + m + n = 0, l^2 + m^2- n^2 = 0.$

Evaluate the following integrals:
$\int\frac{\cos^3\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Using integration, find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x = 4.

Verify Rolle's theorem for the following function on the indicated intervals $f(x) = x(x^{ }- 4)^2$ on the interval $[0, 4]$

Prove that $\begin{vmatrix} \text{yz -x}^{2} & \text{zx - y}^{2} & \text{xy - z}^{2} \\ \text{zx - y}^{2} & \text{xy - z}^{2} & \text{yz - x}^{2} \\ \text{xy - z}^{2} & \text{yz - x}^{2} & \text{zx - y}^{2} \end{vmatrix}$ is divisible by (x + y + z), and hence find the quotient.

Find the shortest distance between the following pairs of parallel lines whose equations are:$\vec{\text{r}}=\big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$ and $\vec{\text{r}}=\big(2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}\big)+\mu\big(-\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)$