Download App

Differential Equations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}-\text{x}\sin^2\text{x}=\frac{1}{\text{x}\log\text{x}}$
✓

Answer

$\frac{\text{dy}}{\text{dx}}-\text{x}\sin^2\text{x}=\frac{1}{\text{x}\log\text{x}}$
$\frac{\text{dy}}{\text{dx}}=\frac{1}{\text{x}\log\text{x}}+\text{x}\sin^2\text{x}$
$\text{dy}=\Big(\frac{1}{\text{x}\log\text{x}}+\text{x}\sin^2\text{x}\Big)\text{dx}$
$\int\text{dy}=\int\frac{1}{\text{x}\log\text{x}}\text{dx}+\int\text{x}\sin^2\text{x dx}$
$\text{y}=\text{I}_1+\text{I}_2$
$\text{I}_1=\int\frac{1}{\text{x}\log\text{x}}\text{ dx}$
Let $\log\text{x}=\text{t}$
$\frac{1}{\text{x}}\ \text{dx}=\text{dt}$
$\text{I}_1=\int\frac{\text{dt}}{\text{t}}$
$=\log|\text{t}|+\text{C}_1$
$\text{I}_1=\log|\log\text{x}|+\text{C}_1$
$\text{I}_2=\int\text{x}\sin^2\text{x dx}$
$=\int\text{x}\frac{(1-\cos2\text{x})}{2}\ \text{dx}$
$=\frac{1}{2}\int(\text{x}-\text{x}\cos2\text{x})\text{dx}$
$=\frac{1}{2}\int\text{x dx}-\frac{1}{2}\int\text{x}\cos2\text{x dx}$
$=\frac{1}{2}\Big(\frac{\text{x}^2}{2}\Big)-\frac{1}{2}[\text{x}\int\cos2\text{x dx}-\int(1\times\int\cos2\text{x dx})\text{dx}]+\text{C}_2$
$=\frac{\text{x}^2}{4}-\frac{1}{2}\Big[\frac{\text{x}\sin\text{x}}{2}+\frac{\cos2\text{x}}{4}\Big]+\text{C}_2$
$\text{I}_2=\frac{\text{x}^2}{4}-\frac{\text{x}\sin2\text{x}}{4}-\frac{\cos2\text{x}}{8}+\text{C}_2$
Put the values of $I_1$ and $I_2$ in equation (1)
$\text{y}=\text{I}_1+\text{I}_2$
$\text{y}=\log|\log\text{x}|+\frac{\text{x}^2}{4}-\frac{\text{x}\sin2\text{x}}{4}-\frac{\cos2\text{x}}{8}+\text{C}\text{ as}\text{ C}_1+\text{C}_2=\text{C}$

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 "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

Draw a rough sketch of the curve $\text{y}=\frac{x}{\pi}+2\sin^2\text{x}$ and find the area between x-axis, the curve and the ordinates $\text{x}=0\text{ and }\text{x}=\pi.$
→
Let n be a fixed positive integer. Define a relation R on Z as follows:
$(\text{a, b})\in\text{R}\Leftrightarrow\ \text{a}-\text{b}$ is divisible by n. Show that R is an equivalence relation on Z.
→
Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-3\frac{\text{dy}}{\text{dx}}+2\text{y}=0,\text{y}(0)=1,\text{y}(0)=3$
Function $\text{y}=\text{e}^\text{x}+\text{e}^{2\text{x}}$
→
If $\tan^{-1}\Big(\frac{\text{x}^2-\text{y}^2}{\text{x}^2+\text{y}^2}\Big)=\text{a}$ prove that $\frac{\text{dx}}{\text{dx}}=\frac{\text{y}}{\text{x}}\frac{(1-\tan\text{a})}{(1+\tan\text{a})}$
→
If $f : R \rightarrow (-1, 1)$ defined by $\text{f(x)}=\frac{10^{\text{x}}-10^{-\text{x}}}{10^{\text{x}}+10^{-\text{x}}}$ is invertible, find $f^{-1}$.
→
Prove that the given vectors are non-coplanar:
$3\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}},\ 2\hat{\text{i}}-\hat{\text{j}}+7\hat{\text{k}}$ and $7\hat{\text{i}}-\hat{\text{j}}+23\hat{\text{k}}$
→
$\int\text{x}^2\sqrt{\text{x}+2}\text{ dx}$
→
$\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ are the position vectors of points A, B and C respectively, prove that:
$\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{c}}\times\vec{\text{a}}$ is a vector perpendicular to the plane of triangle ABC.
→
Evaluate the following integrals:$\int_{0}^\limits{\frac{\pi}{2}}\frac{\sin\text{x}\cos\text{x}}{1+\sin^4\text{x}}\text{ dx}$
→
Solve the following differential equation:
$(\text{x}+2\text{y})\text{dx}-(2\text{x}-\text{y})\text{dy}=0$
→