Download App

Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
For the differential equaton $\text{xy}\frac{\text{dy}}{\text{dx}}=(\text{x}+2)(\text{y}+2).$ Find the solution curve passing through the point (1, -1).
✓

Answer

We have,
$\text{xy}\frac{\text{dy}}{\text{dx}}=(\text{x}+2)(\text{y}+2)$
$\Rightarrow\frac{\text{y}}{\text{y}+2}\text{dy}=\frac{(\text{x}+2)}{\text{x}}\text{dx}$
Integrating both sides, we get
$\int\frac{\text{y}}{\text{y}+2}\text{dy}=\int\frac{(\text{x}+2)}{\text{x}}\text{dx}$
$\Rightarrow\int\text{dy}-2\int\frac{1}{\text{y}+2}\text{dy}=\int\text{dx}+2\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\text{y}-2\log|\text{y}+2|=\text{x}+2\log|\text{x}|+\text{C}...(1)$
This equation represents the family of solution curves of the given differential equation.
We have to find a particular member of the family, which passes through the point (1, -1).
Substituting $\text{x}=1$ and $\text{y}=-1$ in (1), we get
$-1-2\log|1|=1+2\log|1|+\text{C}$
$\Rightarrow\text{C}=-2$
Putting $\text{C}=-2$ in (1), we get
$\text{y}-2\log|\text{y}+2|=\text{x}+2\log|\text{x}|-2$
$\Rightarrow\text{y}-\text{x}+2=\log\Big\{\text{x}^2(\text{y}+2)^2\Big\}$
Hence, $\text{y}-\text{x}+2=\log\Big\{\text{x}^2(\text{y}+2)^2\Big\}$ is the equation of the required curve.

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 $\sqrt{\text{y}+\text{x}}+\sqrt{\text{y}-\text{x}}=\text{c},$ show that $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}-\sqrt{\frac{\text{y}^2}{\text{x}^2}-1}$
→
A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines, and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at ₹ 7 profit and that of B at a profit of ₹ 4. Find the production level per day for maximum profit graphically.
→
If $\text{A}=\begin{bmatrix}2&3\\-1&0\end{bmatrix},$ show that $A^2 - 2A + 3I_2 = 0$.
→
Differentiate the following functions with respect to x:
$\cos^{-1}\Big(\frac{1-\text{x}^{2\text{n}}}{1+\text{x}^{2\text{n}}}\Big), <\text{x}<\infty$
→
The pressure p and the volume v of a gas are connected by the relation pv1.4 = const. Find the percentage error in p corresponding to a decrease of 1/2% in v.
→
In a hockey match, both teams A and B scored same number of goals upto the end of the game, so to decide the winner, the refree asked both the captains to throw a die alternately and decide that the team, whose captain gets a first six, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the refree was fair or not.
→
Differentiate the following functions with respect to x:
$(\cos\text{x})^\text{x}+(\sin\text{x})^\frac{1}{\text{x}}$
→
Solve the following system of equations by matrix method:
$3x + y = 7$
$5x + 3y = 12$
→
Evaluate the following intregals:
$\int\frac{1}{4\cos\text{x}-1}\ \text{dx}$
→
Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}|\text{x}|+3,&\text{if }\text{ x}\geq-3\\-2\text{x},&\text{if }-3<\text{ x}<3\\6\text{x}+2,&\text{if }\text{ x}>3\end{cases}$
→