Question

If $\varphi \left(x\right)=\int {\left\{\varphi \left(x\right)\right\}}^{-2}dx$ and $\varphi \left(1\right)=0$ then $\varphi \left(x\right)=$

• A. ${\left\{2\left(x-1\right)\right\}}^{\frac{1}{4}}$
• B. ${\left\{5\left(x-2\right)\right\}}^{\frac{1}{5}}$
• C. ${\left\{3\left(x-1\right)\right\}}^{\frac{1}{3}}$
• D. none of these

Answer 1

The correct option is C ${\left\{3\left(x-1\right)\right\}}^{\frac{1}{3}}$

$\varphi \left(x\right)=\int {\left\{\varphi \right(x\left)\right\}}^{-2}dx$

differentiate Both side, $\varphi \left(x\right)=y$

$y=\int \alpha {\left\{y\right\}}^{-2}dx$

$\frac{dy}{dx}=y^{-2}$

$y^{2}dy=dx$

Integrate both sides (Indefinite)

$\int y^{2}dy=\int dx$

$\Rightarrow \frac{y^3}{3}=x+c$

given at $x=1,y=0$

$0=1+c\Rightarrow c=-1$

$y^3=3(x-1)$

$y={\left\{3\right(x-1\left)\right\}}^{1/3}$