Download App

Differential Equations — Maths STD 12 Science — Question

Maharashtra 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 "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

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\sin^4\text{x}+\cos^4\text{x}\text{ on }\Big[0,\frac{\pi}{2}\Big]$
→
Evaluate the following integrals:$\int\limits^{\pi}_0\frac{\text{x}}{1+\cos\alpha\sin\text{x}}\text{ dx},0<\alpha<\pi$
→
A merchant plans to sell two types of personal computers a desktop model and a portable model that will cost Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakhs and his profit on the desktop model is Rs. 4500 and on the portable model is Rs. 5000.
→
Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X.
→
Evaluate the following intregals:
$\int\frac{1}{5-4\sin\text{x}}\ \text{dx}$
→
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.
→
For the matrix $\text{A}=\begin{bmatrix}1 & -1 & 1 \\2 & 3 & 0 \\ 18 & 2 & 10 \end{bmatrix},$ show that A (adjoint A) = 0.
→
Differentiate the following functions with respect to x:
$\sin^{-1}\Big\{\sqrt{\frac{1-\text{x}}{2}}\Big\},0<\text{x}<1$
→
Solve the following differential equation:$\frac{\text{dy}}{\text{dx}}+\frac{4\text{x}}{\text{x}^2+1}\text{y}+\frac{1}{(\text{x}^2+1)^2}=0$
→
Prove that the points $\hat{\text{i}}-\hat{\text{j}},\ 4\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}$ and $2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}$ are the vertices of a right-angled triangle.
→