Question

The solution of the differential equation $\frac{dy}{dx}=xy+x+y+1$ is

• A. $y=\frac{1}{2}{x}^2+x+c$
• B. $\log \left(y+1\right)=\frac{1}{2}{x}^2+x+c$
• C. $c\left(y+1\right)=e^{x^2+x}$
• D. None of these.

Answer 1

The correct option is B

$\log \left(y+1\right)=\frac{1}{2}{x}^2+x+c$

Explanation of the correct option.

Compute the required value.

Given : $\frac{dy}{dx}=xy+x+y+1$

$\Rightarrow$ $\frac{dy}{dx}=x(y+1)+y+1$

$\Rightarrow$ $\frac{dy}{dx}=(x+1)(y+1)$

$\Rightarrow$ $\frac{dy}{y+1}=(x+1)dx$

Now, Integrate both side,

$\Rightarrow$$\log \left(y+1\right)=\frac{x^2}{2}+x+c$ $\left[\int \frac{1}{x+a}dx=\log \left(x+a\right),\int x^{n}dx=x^{n+1}\right]$

Hence option $\left(B\right)$ is the correct option.