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

If $\text{x}=\text{a}(1+\cos\theta),\text{y}=\text{a}(\theta+\sin\theta),$ prove that
→
Evaluate the following integrals:
$\int\frac{2\text{x}}{\text{x}^3-1}\ \text{dx}$
→
verify that $\text{y}=\log(\text{x}+\sqrt{\text{x}^2+\text{a}^2})^2$ is a solution of the differential equation $(\text{a}^2+\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}=0$
→
Determine the product and use it to $\begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix} \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix}$solve the system of equations x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1.
→
Differentiate the following functions with respect to x:
$\log_\text{x}3$
→
If $\text{A}=\frac{1}{9}\begin{bmatrix}-8 & 1 & 4\\4 & 4 & 7 \\ 1 & -8 & 4 \end{bmatrix},$ prove that $A^{-1} = A^3$.
→
In a class, $5\%$ of the boys and $10\%$ of the girls have an IQ of more than $150$. In this class, $60\%$ of the students are boys. If a student is selected at random and found to have an IQ of more than $150$, find the probability that the student is a boy.
→
The probability that a student entering a university will graduate is 0.4. Find the probability that out of 3 students of the university.
  1. none will graduate.
  2. only one will graduate.
  3. all will graduate.
→
Prove that $\Big(\frac{\text{x}}{\text{a}}\Big)^\text{n}+\Big(\frac{\text{y}}{\text{b}}\Big)^\text{n}=2$ touches the straight line $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=2$ for all n $\in$ N, at all the point (a, b).
→
Differentiate $(\log\text{x})^\text{x}$ with respect to x.
→