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 "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

If $\vec{\alpha}=3\hat{\text{i}}+4\hat{\text{j}}+5\hat{\text{k}}$ and $\vec{\beta}=2\hat{\text{i}}+\hat{\text{j}}-4\hat{\text{k}},$ then express $\vec{\beta}$ in the form of $\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2,$ where $\vec{\beta}_1$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular to $\vec{\alpha}$.

$\int\text{x}(1-\text{x})^{23}\text{dx}$

Find an equation for the set all points that are equidistant from the planes $3x - 4y + 12z = 6$ and $4x + 3z = 7$

If $\text{f}(\text{a}+\text{b}-\text{x})=\text{f(x)},$ then prove that $\int\limits^{\text{b}}_\text{a}\text{x}\text{f(x)}\text{dx}=\frac{\text{a}+\text{b}}{2}\int\limits^{\text{b}}_\text{a}\text{f(x)}\text{dx}$

If the value of c prescribed bye Lagrange's mean value theorem for the function
$\text{f}(\text{x})=\sqrt{\text{x}^2-4}$ defined on $[2, 3].$

Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)$\text{x}\frac{\text{dy}}{\text{dx}}+2\text{y}=\text{x}^2\log\text{x}$

Find the vector and cartesian equation of the line through the point (5, 2, -4) and which is parallel to the vector $3\hat{\text{i}}+2\hat{\text{j}}-8\hat{\text{k}}.$

A die is tossed twice. A 'success' is getting an odd number on a toss. Find the variance of the number of successes.

A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls type B is at most half of that for dolls of types A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs. 12 and Rs. 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximize the profit ?

Prove that:
$\begin{vmatrix}1&\text{b}+\text{c}&\text{b}^2+\text{c}^2\\1&\text{c}+\text{a}&\text{c}^2+\text{a}^2\\1&\text{a}+\text{b}&\text{a}^2+\text{b}^2 \end{vmatrix}=(\text{a}+\text{b})(\text{b}-\text{c})(\text{c}-\text{a})$