Question

If $\varphi \left(x\right)={\varphi }^{\prime }\left(x\right)$ and $\varphi \left(1\right)=2$ then $\varphi \left(3\right)$ is equal to

• A. ${\varphi }^2$
• B. $2{\varphi }^2$
• C. $3{\varphi }^2$
• D. $2{\varphi }^3$

Answer 1

Answer: (A)

${\varphi }^2$

$\varphi \left(x\right)={\varphi }^{\prime }\left(x\right)\Rightarrow \varphi \left(x\right)=\frac{2\varphi \left(x\right)}{dx}$

$\Rightarrow$ $\int dx=\int \frac{d\left(\varphi \right(x\left)\right)}{\varphi \left(x\right)}$

$\Rightarrow x+c=ln\varphi \left(x\right)\Rightarrow \varphi \left(x\right)=k.e^x$

$\varphi \left(1\right)=2\Rightarrow 2=k.e^1$ $\Rightarrow k=2e^{-1}$

$\therefore \varphi \left(x\right)=2.e^{x-1}$

$\varphi \left(3\right)=2.e^2={\varphi }^2$

$\varphi \left(3\right)={\varphi }^2$