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

Question

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

✓

Answer

We have,
$\frac{\text{dy}}{\text{dx}}+\text{y}=\sin\text{x}\ \dots(1)$
Clearly, it is a linear differential equation of the form
$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
where
$\text{P}=1$
$\text{Q}=\sin\text{x}$
$\therefore\ \text{I.F.}=\text{e}^{\int\text{Pdx}}=\text{e}^{\int\text{dx}}=\text{e}^{\text{x}}$
Multiplying both sides of (1) by $e^x$, we get
$\text{e}^\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}\Big)=\text{e}^\text{x}\sin \text{x}$
$\Rightarrow\ \text{e}^\text{x}\frac{\text{dy}}{\text{dx}}+\text{e}^{\text{x}}\text{y}=\text{e}^\text{x}\sin\text{x}$
Integrating both sides with respect to x, we get
$\text{y}\text{e}^{\text{x}}=\int\text{e}^{\text{x}}\sin\text{x dx + C}$
$\Rightarrow\ \text{y}\text{e}^{\text{x}}=\frac{\text{e}^{\text{x}}}{2}(\sin\text{x}-\cos{\text{x}})+\text{C}$
$\Rightarrow\ \text{y}=\text{Ce}^{-\text{x}}+\frac{1}{2}(\sin\text{x}-\cos{\text{x}})$
Hence, $\text{y}=\text{Ce}^{-\text{x}}+\frac{1}{2}(\sin\text{x}-\cos{\text{x}})$ 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 "Solve the Following Question.(5 Marks)" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following intregals:
$\int\frac{1}{\sin\text{x}-\sqrt{3}\cos\text{x}}\ \text{dx}$

Write the following in the simplest form:
$\tan^{-1}\Big\{\frac{\sqrt{1+\text{x}^2}-1}{\text{x}}\Big\},\text{x}\neq0$

Evaluate the following integrals:$\int^\limits{(\pi)^\frac{2}{3}}_{0}\sqrt{\text{x}}\cos^2\text{x}^{\frac{3}{2}}\text{ dx}$

A factory has three machines $\mathrm{X}, \mathrm{Y}$ and Z producing 1000,2000 and 3000 bolts per day respectively. The machine X produces $1 \%$ defective bolts, $Y$ produces $1.5 \%$ and $Z$ produces $2 \%$ defective bolts. At the end of a day, a bolt is drawn at random and is found to be defective. What is the probability that this defective bolt has been produced by machine X?

If $\text{A}=\begin{bmatrix}1&-2&0\\ 2&1&3\\ 0&-2&1\end{bmatrix}$, find $A^{-1}.$ Using $A^{-1},$ solve the system of linear equations:
$x - 2y = 10, 2x + y + 3z = 8, -2y + z = 7$

Evaluate the following integrals:
$\int\frac{1}{1+\sqrt{\text{x}}}\text{dx}$

Find the angle between the lines whose direction cosines are given by the equations:
$2l + 2m - n = 0, mn + ln + lm = 0$

Evaluate the following integrals as limit of sum:$\int\limits^2_{0}\text{e}^{\text{x}}\text{ dx}$

If $\begin{vmatrix}\text{a}&\text{b}-\text{y}&\text{c}-\text{z}\\\text{a}-\text{x}&\text{b}&\text{c}-\text{z}\\\text{a}-\text{x}&\text{b}-\text{y}&\text{c}\end{vmatrix}=0,$ then using properties of determinants, find the value of $\frac{\text{a}}{\text{x}}+\frac{\text{b}}{\text{y}}+\frac{\text{c}}{\text{z}},$ where $\text{x},\text{y},\text{z}\neq0.$

Prove that:
$\begin{vmatrix}-\text{bc}&\text{b}^2+\text{bc}&\text{c}^2+\text{bc}\\\text{a}^2+\text{ac}&-\text{ac}&\text{c}^2+\text{ac}\\\text{a}^2+\text{ab}&\text{b}^2+\text{ab}&-\text{ab}\end{vmatrix}$
$=(\text{ab}+\text{bc}+\text{ca})^3$