Download App

Differential Equations — Maths STD 12 Science — Question

Maharashtra 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 "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 the plane vector equation is $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}})=1$ and the line whose vector equation is $\vec{\text{r}}=(-\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})+\lambda(2\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}})$ are parallel. Also, find the distance between them.
→
Maximize : z = 11x + 8y subject to x ≤ 4, y ≤ 6, x + y ≤ 6, x ≥ 0, y ≥ 0.
→
Find the points on the curve $y = x^3 - 3x$, where the tangent to the curve is parallel to the chord joining $(1, -2)$ and $(2, 2).$
→
Evaluate the following integrals:$\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\sin\text{x}|\sin\text{x}|\text{dx}$
→
Evaluate the following intregals:
$\int\frac{\cos\text{x}}{\cos3\text{x}}\ \text{dx}$
→
Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)$\text{e}^{-\text{y}}\sec^2\text{y dy}=\text{dx}+\text{x dy}$
→
Find $\frac{d^2 y}{d x^2}$ of the following:

x = a(θ – sin θ), y = a (1 – cos θ)

→
Evaluate the following integrals as limit of sum:$\int\limits^2_{1}\big(\text{x}^2-1\big)\text{dx}$
→
Differentiate the following functions with respect to x:
$\sin^{-1}\Big\{\frac{\sin\text{x}+\cos\text{x}}{\sqrt{2}}\Big\},-\frac{3\pi}{4}<\text{x}<\frac{\pi}{4}$
→
Differentiate the following w. r. t. x.$\frac{e^{x^2}(\tan x)^{\frac{x}{2}}}{\left(1+x^2\right)^{\frac{3}{2}} \cos ^3 x}$
→