Download App

Differential Equations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equation
$(\text{x}^3+\text{x}^2+\text{x}+1)\frac{\text{dy}}{\text{dx}}=2\text{x}^2+\text{x}$
✓

Answer

We have,
$(\text{x}^3+\text{x}^2+\text{x}+1)\frac{\text{dy}}{\text{dx}}=2\text{x}^2+\text{x}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{2\text{x}^2+\text{x}}{\text{x}^3+\text{x}^2+\text{x}+1}$
$\Rightarrow\text{dy}=\frac{2\text{x}^2+\text{x}}{(\text{x}+1)(\text{x}^2+1)}\ \text{dx}$
Intregrating both sides, we get
$\int\text{dy}=\int\Big\{\frac{2\text{x}^2+\text{x}}{(\text{x}+1)(\text{x}^2+1)}\Big\}\text{dx}$
$\Rightarrow\text{y}=\int\Big\{\frac{2\text{x}^2+\text{x}}{(\text{x}+1)(\text{x}^2+1)}\Big\}\text{dx}$
Let $\frac{2\text{x}^2+\text{x}}{(\text{x}+1)(\text{x}^2+1)}=\frac{\text{A}}{\text{x}+1}+\frac{\text{Bx}+\text{C}}{\text{x}^2+1}$
$\Rightarrow 2x^2 + x = Ax^2 + A + Bx^2 + Bx + Cx + C$
$\Rightarrow 2x^2 + x = (A + B)x^2+ (B + C)x + (A + C)$
Compairing the coefficient on both sides, we get
$A + B = 2 ...(1)$
$B + C = 1 ...(2)$
$A + C = 0 ...(3)$
Solving (1), (2) and (3), we get
$\text{A}=\frac{1}{2}$
$\text{B}=\frac{3}{2}$
$\text{C}=-\frac{1}{2}$
$\therefore\text{y}=\frac{1}{2}\int\frac{1}{(\text{x}+1)}\ \text{dx}+\int\frac{\frac{3}{2}\text{x}-\frac{1}{2}}{\text{x}^2+1}\ \text{dx}$
$=\frac{1}{2}\int\frac{1}{(\text{x}+1)}\text{dx}+\frac{3}{4}\int\frac{2\text{x}}{\text{x}^2+1}\ \text{dx}-\frac{1}{2}\int\frac{1}{\text{x}^2+1}\ \text{dx}$
$=\frac{1}{2}\log|\text{x}+1|+\frac{3}{4}\log|\text{x}^2+1|-\frac{1}{2}\tan^{-1}\text{x}+\text{C}$
Hence, $\text{y}=\frac{1}{2}\log|\text{x}+1|+\frac{3}{4}\log|\text{x}^2+1|-\frac{1}{2}\tan^{-1}\text{x}+\text{C}$ is the solution to the given differential equation.

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:$\int_{0}^\limits{1}\frac{24\text{x}^3}{(1+\text{x}^2)^4}\text{ dx}$
→
Evaluate the following integrals as limit of sum:$\int\limits^1_{-1}(\text{x}+3)\text{dx}$
→
Evaluate the following integrals:
$\int\frac{1}{4\text{x}^2+12\text{x}+5}\text{dx}$
→
Discuss the continuity of the following functions at the indicated point:
$\text{f(x)}=\begin{cases}\frac{\text{x}^\text{x}-1}{\log(1+2\text{x})}&,\text{ if x}\neq0&\\ \ &&\text{at x}=0\\7&,\text{ if x}=0\end{cases}$
→
If $\text{y}=\text{x}\sin^{-1}\text{x}+\sqrt{1-\text{x}^2},$ prove that $\frac{\text{dy}}{\text{dx}}=\sin^{-1}\text{x}$
→
Find the equation of the perpendicular drawn from the point P(-1, 3, 2) to the line $\vec{\text{r}}=\big(2\hat{\text{j}}+3\hat{\text{k}}\big)+\lambda\big(2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}}\big).$ Also, find the coordinates of the foot of the perpendicular from P.
→
Find the angle between the pairs of lines with direction ratios proportional to
$2, 2, -2$ and $4, 1, 8$
→
Find the area of the region $\{(\text{x},\text{y}):\text{x}^2+\text{y}^2\leq4,\text{x}+\text{y}\geq2\}$
→
Find the particular solution of the following differential equations:

$2 e^{\frac{x}{y}} d x+\left(y-2 x e^{\frac{x}{y}}\right) d y=0$, when $y(0)=1$

→
Prove that:
$\begin{vmatrix}\text{a}+\text{b}+2\text{c}&\text{a}&\text{b}\\\text{c}&\text{b}+\text{c}+2\text{a}&\text{b}\\\text{c}&\text{a}&\text{c}+\text{a}+2\text{b} \end{vmatrix}=2(\text{a}+\text{b}+\text{c})^3$
→