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Types of solutions of differential equations

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Question

$The solution of the differential equation \frac{d y}{d x} - y = 1 - e^{- x} and y (0) = y, has a finite$ $value when x \to \infty then the value of |\frac{2}{y_{0}}| is$

Moderate

Solution

$\frac{d y}{d x} - y = 1 - e^{- x}, I . F = e^{- x}$
$\begin{array}{l} Solution of the D.E is {ye}^{- x} = \int e^{- x} (1 - e^{- x}) dx = \int (e^{^{- x}} - e^{- 2 x}) dx \\ y = - e^{- x} + \frac{e^{- 2 x}}{2} + c at x = 0 \\ y = - 1 + \frac{1}{2} + c y = - \frac{1}{2} + c \Rightarrow c = y + \frac{1}{2} \\ ∴ {ye}^{- x} = - e^{- x} + \frac{e^{- 2 x}}{2} + y + \frac{1}{2}, if x \to \infty \\ \to 0 = 0 + 0 + y_{0} + \frac{1}{2} \to y_{0} = - \frac{1}{2} \\ \Rightarrow |y_{0}| = 0.5 \end{array}$

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