Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS4 Marks
Question
Solve $\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}(\log{\text{y}-\log\text{x}+1}).$
✓

Answer

Given, $\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}(\log{\text{y}-\log\text{x}+1})$
$\Rightarrow \text{x}\frac{\text{dy}}{\text{dx}}=\text{y}\log\Big(\frac{\text{y}}{\text{x}} + 1\Big)$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}\Big(\log\frac{\text{y}}{\text{x}} + 1\Big)\ .....(\text{i})$
Which is a homogeneous equation.
Put $\frac{\text{y}}{\text{x}}=\text{v}$ or $\text{y}=\text{vx}$
$\therefore\frac{\text{dy}}{\text{dx}}=\text{v}+\text{x}\frac{\text{dv}}{\text{dx}}$
On substituting these values in Eq. (i), we get
$\text{v}+\text{x}\frac{\text{dv}}{\text{dx}}=\text{v}(\log\text{v}+1)$
$\Rightarrow\text{x}\frac{\text{dv}}{\text{dx}}=\text{v}(\log\text{v}+1-1)$
$\Rightarrow\text{x}\frac{\text{dv}}{\text{dx}}=\text{v}(\log\text{v})$
$\Rightarrow\frac{\text{dv}}{\text{v}\log\text{v}}=\frac{\text{dx}}{\text{x}}$
On integrating both sides, we get
$\Rightarrow\int\frac{\text{dv}}{\text{v}\log\text{v}}=\int\frac{\text{dx}}{\text{x}}$
On putting $\log\text{v}=\text{u}$ in LHS integral, we get
$\frac{1}{\text{v}}.\text{dv}=\text{du}$
$\int\frac{\text{du}}{\text{u}}=\int\frac{\text{dx}}{\text{x}}$
$\Rightarrow\log\text{u}=\log\text{x}+\log\text{C}$
$\Rightarrow\log\text{u}=\log\text{C}\text{x}$
$\Rightarrow\text{u}=\text{C}\text{x}$
$\Rightarrow\log\text{v}=\text{C}\text{x}$
$\Rightarrow\log\Big(\frac{\text{y}}{\text{x}}\Big)=\text{C}\text{x}$

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

Evaluate the following intregals:
$\int\frac{2\text{x}+5}{\sqrt{\text{x}^2+2\text{x}+5}}\text{dx}$
→
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\frac{(\text{x}^2-1)^3(2\text{x}-1)}{\sqrt{(\text{x}-3)(4\text{x}-1)}}$
→
If the derivative of tan-1 (a + bx) takes the value 1 at x = 0, prove that 1 + a2 = b.
→
Solve the following systems of linear equations by cramer's rule:
3x + y = 19,
3x - y = 23
→
Find the direction cosines of the lines, connected by the relations: l + m + n = 0 and $\frac{2}{\text{m}}+\frac{2}{\text{n}}-\text{mn}=0$.
→
If $\cos\text{y}=\text{x}\cos(\text{a}+\text{y}),$ where $\cos\text{a}\neq\pm1,$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\cos^2(\text{a}+\text{y})}{\sin\text{a}}$
→
Solve the following differential equation:

$\frac{\text{dy}}{\text{dx}}+\text{y}=\text{e}^{-2\text{x}}$

→
Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin 3 times and notes the number of heads. If she gets 1, 2, 3 or 4 she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?
→
$\text{Evaluate:}\int\frac{\text{x + 2}}{\sqrt{\text{x}^{2}+\text{2x}+}\text{3}}\text{dx}$
→
Evaluate the following:
$\begin{bmatrix}1&-1\\0&2\\2&3\end{bmatrix}\begin{pmatrix}\begin{bmatrix}1&0&2\\2&0&1\end{bmatrix}-\begin{bmatrix}0&1&2\\1&0&2 \end{bmatrix}\end{pmatrix}$
→