Question

Solve the initial value problem y′−ay=0,y(x0)=y0

Answer 1

Answer on Question #69628 – Math – Differential Equations

Question

Solve the initial value problem $y' - ay = 0$, $y(x_0) = y_0$

Solution

$$y' - ay = 0, \quad (1)$$ $$y(x_0) = y_0. \quad (2)$$

Equation (1) is a first-order linear ordinary differential equation, its general solution is given by

$$y = C \cdot e^{ax}, \quad (3)$$

where $C$ is an arbitrary real constant.

Substituting for $y$ from (3) into (2)

$$y(x_0) = C \cdot e^{ax0} = y_0 \quad \Rightarrow \quad C = y_0 \cdot e^{-ax0} \quad \Rightarrow \quad y = y_0 \cdot e^{-ax0} \cdot e^{ax} = y_0 \cdot e^{a(x \cdot x_0)}$$

Answer: $y = y_0 \cdot e^{a(x \cdot x_0)}$

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