Question

The solution of the differential equation $\frac{dy}{dx}={(4x+y+1)}^2$ is

• A. $(4x+y+1)=\tan (2x+C)$
• B. ${(4x+y+1)}^2=2\tan (2x+C)$
• C. ${(4x+y+1)}^3=3\tan (2x+C)$
• D. $(4x+y+1)=2\tan (2x+C)$

Answer 1

The correct option is D

$(4x+y+1)=2\tan (2x+C)$

Explanation for correct option

Given: $\frac{dy}{dx}={(4x+y+1)}^2$

Let $v=4x+y+1$

differentiating both sides

$$\begin{gathered} \Rightarrow \frac{dv}{dx}=4+\frac{dy}{dx} \\ \Rightarrow \frac{dv}{dx}=4+{\left(4x+y+1\right)}^2 \\ \Rightarrow \frac{dv}{dx}=4+{\left(v\right)}^2 \\ \Rightarrow \frac{dv}{v^2+4}=dx \end{gathered}$$

integrating both sides

$$\begin{gathered} \Rightarrow \int \frac{1}{v^2+4}dv=\int dx \\ \Rightarrow \frac{1}{2}{\tan }^{-1}\left(\frac{v}{2}\right)=x+c \\ \Rightarrow {\tan }^{-1}\left(\frac{4x+y+1}{2}\right)=2x+C \\ \Rightarrow \frac{4x+y+1}{2}=\tan (2x+C) \\ \Rightarrow 4x+y+1=2\tan (2x+C) \end{gathered}$$

Hence, option D is correct.