📂Odinary Differential Equations

지수성장방정식/상수 계수를 갖는 1계 선형 동차 미분 방정식

Equation

$$\dfrac{dy}{dx} = \alpha y \tag{1}$$

The general solution of the first-order linear homogeneous differential equation with constant coefficients, as shown above, is as follows:

$$y=Ae^{\alpha x}$$

In this instance, $A$ is a constant.

Explanation

This is also referred to as an exponential growth equation because the solution is an exponential function and it is used to model phenomena such as population growth.

If you consider a function that remains the same even after differentiating once, you can understand why the exponential function is the solution.

Solution

By applying separation of variables to $(1)$, we obtain:

$$\dfrac{dy}{dx} = \alpha y \implies \dfrac{1}{y} dy = \alpha dx$$

After integrating both sides, by the differentiation of the logarithm function, we have:

$$\ln y = a x + C$$

At this point, $C$ is the constant of integration. Finally, by applying the exponential function to both sides, we get:

$$y=e^{\alpha x + C}=e^{\alpha x} e^{C}=Ae^{\alpha x}$$

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