First slide

Methods of solving first order first degree differential equations

Question

$Which of the following is true for y (x) that satisfies the differential equation$ $\frac{d y}{d x} = x y - 1 + x - y; y (0) = 0$

Moderate

A

$y (1) = e^{- \frac{1}{2}} - 1$
B

$y (1) = 1$
C

$y (1) = e^{- \frac{1}{2}} - e^{- \frac{1}{2}}$
D

$y (1) = e^{\frac{1}{2}} - 1$

Solution

$\begin{array}{l} \frac{d y}{d x} = x y - 1 + x - y & \frac{d y}{d x} = (x - 1) (y + 1) \\ \int \frac{d y}{y + 1} = \int (x - 1) d x & \log (y + 1) = \frac{x^{2}}{2} - x + c \to (1) \\ y (0) = 0 \Rightarrow c = 0 \\ Sub x = 1 in (1) \end{array}$
$\begin{array}{l} \log (y + 1) = \frac{1}{2} - 1 \\ y + 1 = e^{- 1 / 2} \Rightarrow y = e^{\frac{- 1}{2}} - 1 \end{array}$

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