General Solution of Differential Equations: y'' - 2y' - 3y 0

In this article, we will explore the general solution of the differential equation y'' - 2y' - 3y 0. We will start by understanding the characteristic equation, which leads us to the roots that will help us derive the general solution. This process is fundamental in solving linear differential equations with constant coefficients.

Characterizing the Differential Equation

The given differential equation is a second-order linear homogeneous differential equation with constant coefficients:

y″ - 2y′ - 3y 0

To find the general solution, we use the characteristic equation, which is obtained by replacing y, y', and y'' with r^2, r, 1 respectively. Thus, the characteristic equation is:

r2 - 2r - 3 0

Solving the Characteristic Equation

The characteristic equation is a quadratic equation, which can be factored or solved using the quadratic formula. Let's factor the equation:

r2 - 2r - 3 left( r - 3 right) left( r 1 right) 0

From the factored form, we can find the roots of the characteristic equation by setting each factor to zero:

r 3r -1

The roots r_1 3 and r_2 -1 are distinct real roots. This means our general solution can be expressed as a linear combination of the solutions corresponding to these roots:

y(t) C_1 e^{r_1 t} C_2 e^{r_2 t}

Substituting the roots:

y(t) C1 e^{3t} C2 e^{-t}

Conclusion

The general solution to the differential equation y'' - 2y' - 3y 0 is:

y(t) C1 e^{3t} C2 e^{-t}

Here, C_1 and C_2 are arbitrary constants determined by initial conditions. This general solution encapsulates the behavior of all possible solutions to the differential equation.

Understanding and solving differential equations using the characteristic equation and its roots is a crucial skill in mathematics and many applied sciences, especially in engineering and physics. It provides a systematic approach to find the solutions to a wide variety of linear differential equations.

Related Keywords

This article is relevant to the following keywords:

differential equations characteristic equation general solution