Download App

DIFFERENTIAL EQUATIONS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDIFFERENTIAL EQUATIONS5 Marks
Question
For each of the differential equations in find a particular solution satisfying the given condition:
$(\text{x}^3+\text{x}^2+\text{x}+1) \frac{\text{dy}}{\text{dx}} = 2\text{x}^2+\text{x; y} =1$  when $ x = 0$
✓

Answer

The given differential equation is$(\text{x}^3+\text{x}^2+\text{x}+1)\frac{\text{dy}}{\text{dx}}=2\text{x}^2+\text{x}$
$\text{or} \ \ \ \frac{\text{dy}}{\text{dx}}=\frac{2\text{x}^2+\text{x}}{\text{x}^3+\text{x}^2+\text{x}+1}$
$\text{or} \ \ \ \frac{\text{dy}}{\text{dx}}=\frac{2\text{x}^2+\text{x}}{\text{x}^2(\text{x}+1)+1(\text{x}+1)}$
$\text{or} \ \ \ \ \frac{\text{dy}}{\text{dx}}=\frac{2\text{x}^2+\text{x}}{(\text{x}+1)(\text{x}^2+1)}$
Separting the variables, we get, $\text{dy}=\frac{2\text{x}^2+\text{x}}{(\text{x}+1)(\text{x}^2+1)}\text{dx}$ Integrating, $\int\text{dy=} \ \int\frac{2\text{x}^2+\text{x}}{(\text{x}+1)(\text{x}^2+1)}\text{dx} \ \ ...(1)$
Put $\frac{2\text{x}^2+\text{x}}{\text{(x}-1)(\text{x}^2+1)}=\frac{\text{A}}{\text{x}+1}+\frac{\text{Bx}+\text{C}}{\text{x}^2+1} \ \ ...(2 )$
$\Rightarrow \ 2\text{x}^2 +\text{x}=\text{Ax}^2 + \text{A} +\text{Bx}^2 +\text{Cx + Bx + C}$
$\Rightarrow \ \ 2\text{x}^2 +\text{x} = (\text{A + B)x}^2+\ (\text{B + C)x + A + C} \ \ \ \ ......(3)$
Now comparing the coefficients of $x^2$ and $x$
$ \Rightarrow \text{A + B} =2$
$\Rightarrow \text{B + C}=1$
$\Rightarrow \text{A + C}=0$
Solving tham we will get the values of $\text{A, B, C} \text{ A}= \frac{1}{2}, \text{B} =\frac{3}{2}-\frac{1}{2}$ Putting the values of $\text{A,B,C} \therefore \text{from}(2),$
$\frac{2\text{x}^2+\text{x}}{(\text{x}+1)(\text{x}^2+1)} = \frac{\frac{1}{2}}{\text{x}+1}+\frac{\frac{3}{2}\text{x}-\frac{1}{2}}{\text{x}^2+1}\text{or}\ \ \frac{2\text{x}^2+\text{x}}{(\text{x}+1)(\text{x}^2+1)} \equiv \frac{1}{2(\text{x}+1)}+\frac{3}{2}\bigg(\frac{x}{x^2+1}\bigg)-\frac{1}{2}\bigg(\frac{1}{x^2+1}\bigg)$
$\therefore \text{from}(1), \ \int\text{dy}=\frac{1}{2} \int\frac{1}{x+1}\text{dx}+\frac{3}{2}\int\frac{\text{x}}{\text{x}^2+1}\text{dx}-\frac{1}{2}\int \frac{1}{\text{x}^2+1}\text{dx}$
$\therefore \ \int 1\ \text{dy}= \frac{1}{2}\int\frac{1}{\text{x}+1}\text{dx}+\frac{3}{4}\int\frac{2\text{x}}{\text{x}^2+1} \text{dx}-\frac{1}{2}\int\frac{1}{\text{x}^2+1}\text{dx}$
$\therefore \ \text{y}= \frac{1}{2}\text{log}|\text{x}+1|+ \frac{3}{4}\text{log}(\text{x}^2+1)-\frac{1}{2}\text{tan}^{-1}\text{x}+\text{c} \ \ \ \ \ .....(4)$
Now $y = 1$ when $x = 0$
$\therefore 1=\frac{1}{2}\text{log}(1)+\frac{3}{4}\text{log} \ 1-\frac{1}{2}\text{tan}^{-1} \text{0+c}$
$\therefore 1 =\frac{1}{2}(0)+\frac{3}{4}(0)-\frac{1}{2}(0)+\text{c} $
$\Rightarrow\ \ \text{c}=1$
$\therefore \text{from}\ (4),$
$\text{y}= \frac{1}{2}\text{log}|\text{x}+1|+\frac{3}{4}\text{log}(\text{x}^2+1)-\frac{1}{2}\text{tan}^{-1}\text{x}+1$

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 the following integrals:
$\int(3\text{x}+1)\sqrt{4-3\text{x}-2\text{x}^2}\text{dx}$
→
If $xy = e^{x-y},$ find $\frac{\text{dy}}{\text{dx}}$
→
$\int\text{e}^{-3\text{x}}\cos^3\text{x dx}$
→
Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\frac{1-\text{t}^2}{1+\text{t}^2}\text{ and y}=\frac{2\text{t}}{1+\text{t}^2}$
→
In a large bulk of items, $5$ percent of the items are defective. What is the probability that a sample of $10$ items will include not more than one defective item?
→
If $\text{xy}=4,$ prove that $\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}^2\Big)=3\text{y}$
→
Maximize Z = 18x + 10y
Subject to
$4\text{x}+\text{y}\geq20$
$2\text{x}+3\text{y}\geq30$
$\text{x},\text{y}\geq0$
→
The vector equations of two lines are:$\overrightarrow{r} = \hat{i} +2\hat{j} +3\hat{k} +\lambda (\hat{i}-3\hat{j} +2\hat{k}) \text{and} \overrightarrow{r} = 4\hat{i} +5\hat{j} +6\hat{k} + \mu (2\hat{i}-3\hat{j} +\hat{k})$
Find the shortest distance between the above lines.
→
If a young man rides his motorcycle at 25 km/hour, he had to spend Rs. 2 per km on petrol. If he rides at a faster speed of 40 km/hour, the petrol cost increases at Rs. 5 per km. He has Rs. 100 to spend on petrol and wishes to find what is the maximum distance he can travel within one hour. Express this as an LPP and solve it graphically.
→
Differentiate the following with respect to x:
$\cos^{-1}\Big(\frac{1-\text{x}}{1+\text{x}}\Big)$
→