Download App

Differential Equations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equations:$(\text{xy}^2+2\text{x})\text{dx}+(\text{x}^2\text{y+2y})\text{dy}=0$
✓

Answer

We have,$(\text{xy}^2+2\text{x})\text{dx}+(\text{x}^2\text{y}+2\text{y})\text{dy}=0$
$\Rightarrow\text{x(y}^2+2)\text{dx+y}(\text{x}^2+2)\text{dy}=0$
$\Rightarrow\text{x(y}^2+2)\text{dx}=-\text{y}(\text{x}^2+2)\text{dy}$
$\Rightarrow\frac{\text{x}}{(\text{x}^2+2)}\text{dx}=-\frac{\text{y}}{(\text{y}^2+2)}\text{dy}$
Integration both sides, we get
$\int\frac{\text{x}}{\text{x}^2+2}\text{dx}=-\int\frac{\text{y}}{\text{y}^2+2}\text{dy}$
$\Rightarrow\frac{1}{2}\int\frac{2\text{x}}{\text{x}^2+2}\text{dx}=-\frac{1}{2}\frac{2\text{y}}{\text{y}^2+2}\text{dy}$
$\Rightarrow\frac{1}{2}\log|\text{x}^2+2|=-\frac{1}{2}\log|\text{y}^2+2|+\log\text{C}$
$\Rightarrow\frac{1}{2}\log|\text{x}^2+2|+\frac{1}{2}\log|\text{y}^2+2|=\log\text{C}$
$\Rightarrow\log|\text{x}^2+2|+\log|\text{y}^2+2|=2\log\text{C}$
$\Rightarrow\log\big(|\text{x}^2+2||\text{y}^2+2|\big)=\log\text{C}^2$
$\Rightarrow\big(|\text{x}^2+2||\text{y}^2+2|\big)=\text{C}^2$
$\Rightarrow(\text{x}^2+2)(\text{y}^2+2)=\text{K}$
$\Rightarrow\text{y}^2+2=\frac{\text{K}}{\text{x}^2+2}$

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

Show that $\text{f}\text{ (x)}=\begin{cases}\frac{\text{|x}-\text{a}|}{|\text{x}-\text{a}|}, & \text{when} \text{ x}\neq 0\\2, & \text{when}\text{ x} = 0\end{cases}$ is discontinuous at x = a.
→
A kite is $120\ m$ high and $130\ m$ of string is out. If the kite is moving away horizontally at the rate of $52m/ sec$, find the rate at which the string is being paid out.
→
Find all point of discontinuity of the function $\text{f(t)}=\frac{1}{\text{t}^2+\text{t}-2},$ where $\text{t}=\frac{1}{\text{x}-1}$
→
$\triangle=\begin{vmatrix}\cos\alpha\cos\beta&\cos\alpha\sin\beta&-\sin\alpha\\-\sin\beta&\cos\beta&0\\\sin\alpha\cos\beta&\sin\alpha\sin\beta&\cos\alpha \end{vmatrix}$
→
Find the angle of intersecting of the following curves:
$\text{x}^2+\text{y}^2-4\text{x}-1=0\text{ and }\text{x}^2+\text{y}^2-2\text{y}-9=0$
→
Suppose a girl throws a die. If she gets $1$ or $2$, she tosses a coin three times and notes the number of tails. If she gets $3, 4, 5$ or $6$, she tosses a coin once and notes whether a 'head' or 'tail' is obtained. If she obtained exactly one 'tail', then what is the probability that she threw $3, 4, 5$ or $6$ with the die?
→
Evaluate :

$\int \frac{3 x+4}{\sqrt{2 x^2+2 x+1}} d x$

→
Evaluate the following intregals:
$\int\frac{3+4\text{x}-\text{x}^2}{(\text{x}+2)(\text{x}-1)}\ \text{dx}$
→
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is $\frac{2\text{R}}{\sqrt{3}}$ .
→
Prove the following results:
$\tan^{-1}\frac{1}{4}+\tan^{-1}\frac{2}{9}=\frac{1}{2}\cos^{-1}\frac{3}{5}=\frac{1}{2}\sin^{-1}\Big(\frac{4}{5}\Big)$
→