Solve the following differential equation: dy dx =(x+y+1)^ 2 - Vidyadip

Question

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=(\text{x}+\text{y}+1)^{2}$

✓

Answer

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

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 "5 Marks Questions" from Differential Equations→Differential Equations — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\limits^{\frac{3}{2}}_0\big|\text{x}\cos\pi\text{x}\big|\text{dx}$

Solve the following differential equation:
$\big(\cot^{-1}\text{y} + \text{x}\big)\text{dy}= \big(1 + \text{y}^2\big) \text{dx}$

Show that $\text{y}=4\text{ax}$ is a solution of the differential equation $\text{y}=\text{x}\frac{\text{d}\text{y}}{\text{dx}}+\text{a}\frac{\text{dx}}{\text{dy}}.$

Find the equation of the plane which contains two parallel lines $\frac{\text{x}-4}{1}=\frac{\text{y}-3}{-4}=\frac{\text{z}-2}{5}$ and $\frac{\text{x}-3}{1}=\frac{\text{y}+2}{-4}=\frac{\text{z}}{5}.$

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\frac{\text{x}}{2}-\sin\frac{\pi\text{x}}{6}\text{ on }[-1,0]$

Find the area of a parallelogram ABCD whose side AB and the diagonal AC are given by the vectors $3\hat{\text{i}} + \hat{\text{j}}+4\hat{\text{k}}$ and $4\hat{\text{i}} + 5\hat{\text{k}} $ respectively.

Show that the function $f: R \rightarrow\{x \in R:-1< x < 1\}$ defined by $f(x)=\frac{x}{1+|x|}, x \in R$ is one$-$one and onto function.

Prove that: If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

Evaluate the following definite integrals:
$\int\limits_{0}^{\frac{\pi}{6}}\cos\text{x }\cos2\text{x}\text{ dx}$

If $\text{y}=\text{e}^{\text{x}}\cos\text{x},$ Prvoe that $\frac{\text{dy}}{\text{dx}}=\sqrt{2}\text{e}^\text{x}.\cos\Big(\text{x}+\frac{\pi}{4}\Big)$