Download App

DIFFERENTIAL EQUATIONS — MATHS STD 12 Science — Question

Rajasthan 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 $(1)$
$\text{y}=\text{I}_1+\text{I}_2$
$\text{y}=\log|\log\text{x}|+\frac{\text{x}^2}{4}-\frac{\text{x}\sin2\text{x}}{4}-\frac{\cos2\text{x}}{8}+\text{C}$ 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 "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

Evaluate: $\int\frac{2}{\text{(1-x)(1+x)}^{2}}\text{dx}$
→
Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$\text{x}^{\frac{2}{3}}+\text{y}^{\frac{2}{3}}=\text{a}^{\frac{2}{3}}$
→
Find the angle of intersecting of the following curves:
$\text{x}^2+\text{y}^2-4\text{x}-1=0\text{ and }\text{x}^2+\text{y}^2-2\text{y}-9=0$
→
Show that the lines $\frac{5-\text{x}}{-4}=\frac{\text{y}-7}{4}=\frac{\text{z}+3}{-5}$ and $\frac{\text{x}-8}{7}=\frac{2\text{y}-8}{2}=\frac{\text{z}-5}{3}$ are coplanar.
→
Evaluate the following integrals:$\int\frac{1}{\sqrt{3\text{x}^2+5\text{x}+7}}\text{ dx}$
→
Find the shortest distance between the lines whose vector equations are:
$\vec{\text{r}}=\Big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\Big)+\lambda\Big(\hat{\text{i}}-3\hat{\text{j}}+2\hat{\text{k}}\Big)$
$\text{and}\ \vec{\text{r}}=4\hat{\text{i}}+5\hat{\text{j}}+6\hat{\text{k}}+\mu\Big(2\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}\Big)$
→
Find the shortest distance between the lines
$\frac{\text{x}+1}{7}=\frac{\text{y}+1}{-6}=\frac{\text{z}+1}{1}$ and $\frac{\text{x}-3}{1}=\frac{\text{y}-5}{-2}=\frac{\text{z}-7}{1}$
→
Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.
→
Find the real values of $\lambda$ for which the following system of linear equations has non-trivial solutions. Also, find the non-trivial solutions:
$2\lambda\text{x}-2\text{y}+3\text{z}=0,$
$\text{x}+\lambda\text{y}+2\text{z}=0,$
$2\text{x}+0\text{y}+\lambda\text{z}=0$
→
If $\text{x}=\sin^{-1}\Big(\frac{2\text{t}}{1+\text{t}^2}\Big)$ and $\text{y}=\tan^{-1}\Big(\frac{2\text{t}}{1-\text{t}^2}\Big),-1<\text{t}<1,$ prove that $\frac{\text{dy}}{\text{dx}}=1$
→