Solve the following differential equation: dy dx = (x+y)^2 - Vidyadip

Question

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}} = (\text{x}+\text{y})^2$

✓

Answer

We have,
$\frac{\text{dy}}{\text{dx}} = (\text{x}+\text{y})^2$
Let $\text{ x} + \text{y} = \text{v}$
$\Rightarrow 1 + \frac{\text{dy}}{\text{dx}} = \frac{\text{dv}}{\text{dx}}$
$\Rightarrow \frac{\text{dy}}{\text{dx}} = \frac{\text{dv}}{\text{dx}} - 1$
$\therefore \frac{\text{dv}}{\text{dx}} - 1 = \text{v}^2$
$\Rightarrow \frac{\text{dv}}{\text{dx}} = \text{v}^2 + 1$
$\Rightarrow \frac{1}{\text{v}^2+1}\text{dv} = \text{dx}$
Integrating both sides, we get
$\int\frac{1}{\text{v}^2+1}\text{dv} = \int\text{dx}$
$\Rightarrow \tan^{-1}\text{v} = \text{x} + \text{C}$
$\Rightarrow \text{v} = \tan(\text{x}+\text{C})$
$\Rightarrow \text{x}+\text{y} = \tan (\text{x}+\text{C})$

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

Prove that |A adj $A| = |A|^n.$

Find the area of the region bounded by the curve $y=x-1$ and $(y-1)^2=4(x+1)$.

$\int_{-1}^1 \frac{x^3+2}{\sqrt{x^2+4}} \cdot d x$

Evaluate the following integrals:$\int_{1}^\limits{2}\frac{1}{\text{x}\big(1+\log\text{x}\big)^2}\text{ dx}$

Differentiate the following functions with respect to x:
$\sin^{-1}\big\{\sqrt{1-\text{x}^2}\big\},0<\text{x}<1$

Express $-\hat{i}-3 \hat{j}+4 \hat{k}$ as linear combination of the vectors $2 \hat{i}+\hat{j}-4 \hat{k} 2 \hat{i}-\hat{j}+3 \hat{k}$and $3 \hat{i}+\hat{j}-2 \hat{k}$.

If $\text{f(x)}=\frac{4\text{x}+3}{6\text{x}-4},\ \text{x}\neq\frac{2}{3},$ show that fof(x) = x for all $\text{x}\neq\frac{2}{3}.$ What is the inverse of f?

Evaluvate the following intregals:
$\int\frac{4\sin\text{x}+5\cos\text{x}}{5\sin\text{x}+4\cos\text{x}}\ \text{dx}$

If $\text{A}=\begin{bmatrix}2&-3&5\\ 3&2&-4\\ 1&1&-2\end{bmatrix}$, find $A^{-1}$ and hence solve the system of linear equations:
$2x - 3y + 5z = 11, 3x + 2y - 4z = -5, x + y + 2z = -3$

In a factory, machine A produces $30 \%$ of the total output, machine B produces $25 \%$ and the machine C produces the remaining output. If defective items produced by machines $A , B$ and C are $1 \%, 1.2 \%, 2 \%$ respectively. Three machines working together produce $10000$ items in a day. An item is drawn at random from a day's output and found to be defective. Find the probability that it was produced by machine B?