First slide

Methods of solving first order first degree differential equations

Question

If $ϕ (x) = \int {(ϕ (x))}^{- 2} d x$ and $ϕ (1) = 0$ then $ϕ (x) =$

Moderate

A

${(2 (x - 1))}^{1 / 4}$
B

${(5 (x - 2))}^{1 / 5}$
C

${(3 (x - 1))}^{1 / 3}$
D

none

Solution

$ϕ (x) = \int {(ϕ (x))}^{- 2} d x$
Differentiate with respect to x
$\begin{array}{l} \Rightarrow ϕ^{1} (x) = [ϕ (x)]^{- 2} \\ \Rightarrow (ϕ (x))^{2} \cdot ϕ^{'} (x) = 1 \\ 3 (ϕ (x))^{2} ϕ^{1} (x) = 3 \\ Integrate, we get \\ [ϕ (x)]^{3} = 3 x + c \\ Since ϕ (1) = 0 \Rightarrow 0 = 3 + c \Rightarrow c = - 3 \\ Then [ϕ (x)]^{3} = 3 x - 3 = 3 (x - 1) \\ Hence ϕ (x) = [3 (x - 1)]^{1 / 3} \end{array}$

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