Download App

Differential Equations — Maths STD 12 Science — Question

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

Answer

We have,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x}}{2\text{y}+\text{x}}$
This is a homogeneous differential equation.
Putting y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$, we get
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{x}}{2\text{vx + x}}$
$\Rightarrow\ \text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{1}{2\text{v}+1}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=\frac{1}{2\text{v}+1}-\text{v}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=\frac{1-2\text{v}^2-\text{v}}{2\text{v}+1}$
$\Rightarrow\ \frac{2\text{v}+1}{1-2\text{v}^2-\text{v}}\text{dv}=\frac{1}{\text{x}}\text{dx}$
Integrating both sides, we get
$\int\frac{2\text{v}+1}{1-2\text{v}^2-\text{v}}\text{dv}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\ \int\frac{2\text{v}+1}{2\text{v}^2+\text{v}-1}\text{dv}=-\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\ \int\frac{2\text{v}+1}{2\text{v}(\text{v}+1)-1(\text{v}+1)}\text{dv}=-\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\ \int\frac{2\text{v}+1}{(2\text{v}-1)(\text{v}+1)}\text{dv}=-\int\frac{1}{\text{x}}\text{dx}\ \dots(1)$
Solving left hand side integral of (1), we get
Using partial fraction,
Let $\frac{2\text{v}+1}{(2\text{v}-1)(\text{v}+1)}=\frac{\text{A}}{(2\text{v}-1)}+\frac{\text{B}}{(\text{v}+1)}$
$\therefore\ \text{A}+2\text{B}=2\ \dots(2)$
And $\text{A}-\text{B}=1\ \dots(3)$
Solving (2) and (3), we get
$\text{A}=\frac{4}3$ and $\text{B}=\frac{1}3$
$\therefore\ \int\frac{2\text{v}+1}{(2\text{v}-1)(\text{v}+1)}\text{dy}=\frac{4}3\int\frac{1}{2\text{v}-1}\text{dv}+\frac{1}3\int\frac{1}{\text{v}+1}\text{dv}$
$=\frac{4}{3\times2}\log|2\text{v}-1|+\frac{1}3\log|\text{v}+1|+\log\text{C}$
From (1), we get
$\frac{2}3\log|2\text{v}-1|+\frac{1}3|\text{v}+1|+\log\text{C}=-\log|\text{x}|+\log\text{C}_1$
$\Rightarrow\ \log\Big\{\big|(2\text{v}-1)^2\big||\text{v}+1|\Big\}=-3\log|\text{x}|+\log\text{C}_2$
$\Rightarrow\ \log\Big\{\big|(2\text{v}-1)^2\big||\text{v}+1|\Big\}=\log\Big|\frac{\text{C}_2^3}{\text{x}^3}\Big|$
$\Rightarrow\ (2\text{v}-1)^2(\text{v}+1)=\frac{\text{C}_2^3}{\text{x}^3}$
Putting $\text{v}=\frac{\text{y}}{\text{x}}$, we get
$\Rightarrow\ \Big(\frac{2\text{y}-\text{x}}{\text{x}}\Big)^2\Big(\frac{\text{y}+\text{x}}{\text{x}}\Big)=\frac{\text{C}_2^3}{\text{x}^3}$
$\Rightarrow\ (\text{x + y})(2\text{y}-\text{x})^2=\text{K}$

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following intregals:
$\int\frac{\text{ax}^2+\text{bx}+\text{c}}{(\text{x}-\text{a})(\text{x}-\text{b})(\text{x}-\text{c})}\ \text{dx},$ where a, b, c are distinct
→
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is $\pi:(\pi+2)$.
→
Prove that the curves $y^2=4 x$ and $x^2=4 y$ divide the area of the square bounded by sides $x=0, x=4, y=4$ and $y =0$ into three equal parts.
→
Find the area of the region in the first quadrant enclosed by x-axis, the line $\text{y}=\sqrt{3\text{x}}$ and the circle $x^2+ y^2 = 16$.
→
Using integration, find the area of the region bounded by the triangle whose vertices are (–1, 2), (1, 5) and (3, 4).
→
If $\text{f}\text{(x)}=\begin{cases}\frac{\text{x}^2-1}{\text{x}-1},& \text{for }\text{ x}\neq1 \\2,&\text{for }\text{ x}=1\end{cases}$ Find whether f (x) is continuous at x = 1.
→
If $\text{A}=\begin{bmatrix}2&-3&5\\ 3&2&-4\\ 1&1&-2\end{bmatrix}$, find $A^{-1}$ and hence solve the system of linear equations:
2x - 3y + 5z = 11, 3x + 2y - 4z = -5, x + y + 2z = -3
→
Show that the relation $R$, defined on the set $A$ of all polygons as $R=\left\{\left(P_1, P_2\right): P_1\right.$ and $P_2$ have same number of sides $\}$, is an equivalence relation. What is the set of all elements in A related to the right angle triangle $T$ with sides 3,4 and 5 ?
→
If $f(x) = x^3 + 4x^2 - x$, find $f(A)$, where $\text{A}=\begin{bmatrix}0&1&2\\2&-3&0\\1&-1&0\end{bmatrix}$
→
If $|\vec{\text{a}}|=13,\big|\vec{\text{b}}\big|=5$ and $\vec{\text{a}}.\vec{\text{b}}=60,$ then find $\big|\vec{\text{a}}\times\vec{\text{b}}\big|.$
→