Question

For solving $\frac{dy}{dx}=4x+y+1=v$, the suitable substitution is

• A. $y=vx$
• B. $y=4x+v$
• C. $y=4x$
• D. $y=4x+1=v$

Answer 1

The correct option is D $y=4x+1=v$

$\frac{dy}{dx}=4x+y+1$

Substitute $4x+y+1=v$. Differentiating w.r.t $x$, we get

$4+\frac{dy}{dx}=\frac{dv}{dx}\Rightarrow \frac{dy}{dx}=\frac{dv}{dx}-4$

$\Rightarrow \frac{dv}{dx}-4=v\Rightarrow \frac{dv}{dx}=4+v$

$\Rightarrow \frac{dv}{4+v}=dx\Rightarrow \int \frac{dv}{4+v}=\int dx$

$\Rightarrow \log (4+v)=x+c$