Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS4 Marks
Question
Solve the following differential equation:
$\text{x}^2\frac{\text{dy}}{\text{dx}}=\text{x}^2-2\text{y}^2+\text{xy}$
✓

Answer

Consider the given differential equation
$\text{x}^2\frac{\text{dy}}{\text{dx}}=\text{x}^2-2\text{y}^2+\text{xy}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{x}^2-2\text{y}^2+\text{xy}}{\text{x}^2}$
This is a homogeneous differential equation.
Substituting y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$, we have
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{x}^2-2\text{v}^2\times\text{x}^2+\text{x}\times\text{v}\times\text{x}}{\text{x}^2}$
$\Rightarrow\ \text{v + x}\frac{\text{dv}}{\text{dx}}=1-2\text{v}^2+\text{v}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=1-2\text{v}^2$
$\Rightarrow\ \frac{\text{dv}}{1-2\text{v}^2}=\frac{\text{dx}}{\text{x}}$
$\Rightarrow\ \frac{\text{dv}}{\text{v}^2-\frac{1}2}=-2\frac{\text{dx}}{\text{x}}$
$\Rightarrow\ \int\frac{\text{dv}}{\big(\frac{1}{\sqrt2}\big)^2-\text{v}^2}=2\int\frac{\text{dx}}{\text{x}}$
$\Rightarrow\ \frac{\sqrt2}2\log\bigg(\frac{\frac{1}{\sqrt2}+\text{v}}{\frac{1}{\sqrt2}-\text{v}}\bigg)=2\log\text{x}+\log\text{C}$
$\Rightarrow\ \frac{1}{\sqrt2}\log\Bigg(\frac{\frac{1}{\sqrt2}+\frac{\text{y}}{\text{x}}}{\frac{1}{\sqrt2}-\frac{\text{y}}{\text{x}}}\Bigg)2\log\text{x}+\log\text{C}$
$\Rightarrow\ \frac{1}{\sqrt2}\log\Big(\frac{\text{x + y}\sqrt2}{\text{x}-\text{y}\sqrt2}\Big)2\log\text{x}+\log\text{C}$
$\Rightarrow\ \frac{1}{\sqrt2}\log\Big(\frac{\text{x + y}\sqrt2}{\text{x}-\text{y}\sqrt2}\Big)\log\text{x}^2+\log\text{C}$
$\Rightarrow\ \log\Big(\frac{\text{x + y}\sqrt2}{\text{x}-\text{y}\sqrt2}\Big)^{\frac{1}{\sqrt2}}=\log\text{Cx}^2$
$\Rightarrow\ \Big(\frac{\text{x + y}\sqrt2}{\text{x}-\text{y}\sqrt2}\Big)^{\frac{1}{\sqrt2}}=\text{Cx}^2$
$\Rightarrow\ \Big(\frac{\text{x + y}\sqrt2}{\text{x}-\text{y}\sqrt2}\Big)=\big(\text{Cx}^2\big)^{\sqrt2}$

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 "4 Marks" 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

Find all points of discontinuity of f, where f is defined by:
$\text f(\text x)=\begin{cases}\left|\text x\right|+3, \text{if x}\leq-3\\-2 \text{x},\text{if}-3 < \text x > 3\\6\text{x}+2,\text{if}\ \text{x}\geq3\end{cases}$
→
If $\text{A}=\begin{bmatrix}1&2&0\\-2&-1&-2\\0&-1&1\end{bmatrix}$, find A-1, solve the system of linear equations x - 2y = 10, 2x - y - z = 8, -2y + z = 7
→
$\int\limits_0^{\pi/4}\Bigg(\sqrt{\text{tan x}}+\sqrt{{\text{cot x}}}\Bigg)\text{ dx}=\sqrt{2}\cdot\frac{\pi}{2}$
→
Prove that:
$\cos^{-1}\frac{12}{13}=\sin^{-1}\frac{3}{5}=\sin^{-1}\frac{3}{5}=\sin^{-1}\frac{56}{65}$
→
Is $|\sin\text{x}|$ differentible? What about $\cos|\text{x}|?$
→
For real numbers x and y, define xRy if $\text{x}-\text{y}+\sqrt{2}$ is an irrational number. Then the relation R is:
  1. Reflexive.
  2. Symmetric.
  3. Transitive.
  4. None of these.
→
Find the general solution of $x \frac{d y}{d x}+2 y=x^{2} \log x$
→
A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?
A = the first throw results in head,
B = the last throw results in tail.
→
Kellogg is a new cereal formed of a mixture of bran and rice that contains at least 88 grams of protein and at least 36 milligrams of iron. Knowing that bran contains 80 grams of protein and 40 milligrams of iron per kilogram, and that rice contains 100 grams of protein and 30 milligrams of iron per kilogram, find the minimum cost of producing this new cereal if bran costs Rs. 5 per kg and rice costs Rs 4 per kg.
→
Find the particular solution of the differential equation
2y ex/y dx + (y – 2x ex/y) dy = 0, given that x = 0 when y = 1.
→