Solve the following differential equations: (y + xy)dx+(x-xy^2)dy… - Vidyadip

Question

Solve the following differential equations:

$(\text{y + xy})\text{dx}+(\text{x}-\text{xy}^2)\text{dy}=0$

✓

Answer

We have,
$(\text{y + xy})\text{dx}+(\text{x}-\text{xy}^2)\text{dy}=0$
$\Rightarrow\text{y}(1+\text{x})\text{dx = x}(\text{y}^2-1)\text{dy}$
$\Rightarrow\frac{1+\text{x}}{\text{x}}\text{dx}=\frac{\text{y}^2-1}{\text{y}}\text{dy}$
Integrating both sides, we get
$\int\frac{1+\text{x}}{\text{x}}\text{dx}=\int\frac{\text{y}^2-1}{\text{y}}\text{dy}$
$\Rightarrow\int\frac{1}{\text{x}}\text{dx}+\int\text{dx}=\int\text{y dy}-\int\frac{1}{\text{y}}\text{dy}$
$\Rightarrow\log|\text{x}|+\text{x}=\frac{\text{y}^2}{2}-\log|\text{y}|+\text{C}$
$\Rightarrow\log|\text{x}|+\text{x}-\frac{\text{y}^2}{2}+\log|\text{y}|=\text{C}$
Hence, $\log|\text{x}|+\text{x}-\frac{\text{y}^2}{2}+\log|\text{y}|=\text{C}$ is the required solution.

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

If the lines $\text{x}=5,\frac{\text{y}}{3-\alpha}=\frac{\text{z}}{-2}$ and $\text{x}=\alpha,\frac{\text{y}}{-1}=\frac{\text{z}}{2-\alpha}$ are coplanar, find the values of $\alpha.$

Minimize $Z = x +2 y$ subject to constraints
$
\begin{array}{l}
2 x+2 y \geq 3 ; \\
x+2 y \geq 6 ; \\
x, y \geq 0
\end{array}
$
using graphical method.

Find the inverse of the matrix (if it exists) given $\left[\begin{array}{ccc}1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1\end{array}\right]$

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes $\vec{\text{r}}\cdot(\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}})=5$ and $\vec{\text{r}}\cdot(3\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})=6.$

A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at Rs. 100 and Rs. 120 per unit respectively, how should he use his resources to maximise the total revenue? Form the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

If $\hat{\text{a}}$ and $\hat{\text{b}}$ are unit vectors inclined at an angle $\theta$, prove that

$\tan\frac{\theta}{2}=\frac{\big|\hat{\text{a}}-\hat{\text{b}}\big|}{\big|\hat{\text{a}}+\hat{\text{b}}\big|}$

Evaluate:
$\int\limits^{1}_{-2} \big|\text{x}^{3} - \text{x}\big| \text{ dx}$

Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=3\hat{\text{i}}+8\hat{\text{j}}+3\hat{\text{k}}+\lambda\big(3\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$ and $\vec{\text{r}}=-3\hat{\text{i}}-7\hat{\text{j}}+6\hat{\text{k}}+\mu\big(-3\hat{\text{i}}+2\hat{\text{j}}+4\hat{\text{k}}\big)$

If $\text{x}=\cos\text{t}(3-2\cos^2\text{t}),\text{y}\sin\text{t}(3-2\sin^2\text{t})$ find the value of $\frac{\text{dy}}{\text{dx}}\text{ at t}=\frac{\pi}{4}$

Find all points of discontinuity of f, where f is defined by:

$\text{f(x)}= \begin{cases}\text{x}^{10} - 1,\ \ \text{if x}\leq 1 \\\text{x}^2,\ \ \ \ \ \ \ \ \ \ \text{if x}>1\end{cases}$