Download App

Differential Equations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+2\text{y}=\sin\text{x}$
✓

Answer

We have,
$\frac{\text{dy}}{\text{dx}}+2\text{y}=\sin\text{x}\ \dots(1)$
Clearly, it is a linear differential equation of the form
$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
where
$\text{P}=1$
and
$\text{Q}=\sin\text{x}$
$\therefore$ I.F. $=\text{e}^{\int\text{Pdx}}$
$=\text{e}^{\int2\text{dx}}$
$=\text{e}^{2\text{x}}$
Multiplying both sides of (1) by I.F. $= e^{2x}$​​​​​​​, we get
$\text{e}^{2\text{x}}\Big(\frac{\text{dy}}{\text{dx}}+2\text{y}\Big)=\text{e}^{2\text{x}}\sin\text{x}$
$\Rightarrow\ \text{e}^{2\text{x}}\frac{\text{dy}}{\text{dx}}+2\text{e}^{2\text{x}}\text{y}=\text{e}^{2\text{x}}\sin\text{x}$
Integrating both sides with respect to x, we get
$\text{ye}^{2\text{x}}=\int\text{e}^{2\text{x}}\sin\text{x dx + C}$
$\Rightarrow\ \text{ye}^{2\text{x}}=\frac{1}5\int[2\text{e}^{2\text{x}}(2\sin\text{x}-\cos\text{x}+\text{e}^{2\text{x}}(2\cos\text{x}+\sin\text{x})]$
Putting $\text{e}^{2\text{x}}(2\sin\text{x}-\cos\text{x})=\text{t}$
$\Rightarrow[2\text{e}^{2\text{x}}(2\sin\text{x}-\cos\text{x})+\text{e}^{2\text{x}}(2\cos\text{x}+\sin\text{x})]\text{dx}=\text{dt}$
$\therefore\ \text{ye}^{2\text{x}}=\frac{1}5\int\text{dt + C}$
$\Rightarrow\text{ye}^{2\text{x}}=\frac{\text{t}}5+\text{C}$
$\text{ye}^{2\text{x}}=\frac{\text{e}^{2\text{x}}}5(2\sin\text{x}-\cos\text{x})+\text{C}$
$\Rightarrow\ \text{y}=\frac{1}5(2\sin\text{x}-\cos\text{x})+\text{Ce}^{-2\text{x}}$
Hence, $\text{y}=\frac{1}5(2\sin\text{x}-\cos\text{x})+\text{Ce}^{-2\text{x}}$ 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 integrals as limit of sum:$\int\limits^{4}_{1}\big(\text{x}^2-\text{x}\big)\text{dx}$
→
Find the angles which the vector $\vec{\text{a}}=\hat{\text{i}}-\hat{\text{j}}+\sqrt{2}\hat{\text{k}}$ makes with the coordinate axes.
→
Prove the following results:
$\sin^{-1}\frac{12}{13}+\cos^{-1}\frac{4}{5}+\tan^{-1}\frac{63}{16}=\pi$
→
Evaluate the following integrals as limit of sum:$\int\limits^{2}_{0}\big(\text{x}^2-\text{x}\big)\text{dx}$
→
Evaluate the following integrals:$\int\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)\text{dx}$
→
If the value of c prescribed bye Lagrange's mean value theorem for the function
$\text{f}(\text{x})=\sqrt{\text{x}^2-4}$ defined on $[2, 3].$
→
Find all point of discontinuity of the function $\text{f(t)}=\frac{1}{\text{t}^2+\text{t}-2},$ where $\text{t}=\frac{1}{\text{x}-1}$
→
Determine the binomial distribution whose mean is 20 and variance 16.
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\cos^3\text{x}\sin^2\text{x}+\text{x}\sqrt{2\text{x}+1}$
→
Evaluate the following integral:
$\int\frac{1}{\sqrt{(2-\text{x})^2-1}}\text{ dx}$
→