Solving Second-Order Linear Differential Equations Using Characteristic Equations
Introduction to Differential Equations
Differential equations are essential in the field of mathematics and physics. They describe the rate of change of a function with respect to one or more independent variables. In this article, we focus on solving a second-order linear differential equation of the form:
Y'' Y' - 2Y 0
This article will guide you through the process of solving this equation using the characteristic equation method. By the end of this article, you'll understand the core concepts and be able to solve similar differential equations.
Understanding the Problem
The given differential equation is:
y'' y' - 2y 0
This is a second-order linear differential equation with constant coefficients. The general form of such an equation is:
y'' p(y) y' q(y) y 0
The Characteristic Equation
To solve this type of equation, we follow a method that involves finding the characteristic equation. We assume a solution of the form:
y Ae^(kx)
Where A and k are constants. We then find the first and second derivatives:
y' kAe^(kx), y'' k^2Ae^(kx)
Substituting these derivatives back into the original differential equation, we get:
k^2Ae^(kx) kAe^(kx) - 2Ae^(kx) 0
Factoring out Ae^(kx), we get:
Ae^(kx)(k^2 k - 2) 0
Since Ae^(kx) is never zero, we have:
k^2 k - 2 0
This is the characteristic equation. We solve this quadratic equation by factoring:
(k 2)(k - 1) 0
Solving for k, we get two roots:
k -2, k 1
Forming the Complementary Function
The complementary function (C.F.) is given by the linear combination of the solutions corresponding to these roots:
C.F. Ae^(-2x) Be^x
Here, A and B are arbitrary constants.
Particular Integral (P.I.)
In this specific problem, we solve for a general solution. Since the right-hand side of the original differential equation is zero, there is no particular integral. Hence, the solution is the same as the complementary function:
Y C.F. P.I.
Since P.I. is zero:
Y Ae^(-2x) Be^x
Conclusion
In summary, we have solved the differential equation y'' y' - 2y 0 using the characteristic equation method. The solution is:
y Ae^(-2x) Be^x
This method can be applied to a wide range of second-order linear differential equations with constant coefficients. The key steps involve forming the characteristic equation, solving it to get the roots, and using these roots to construct the complementary function and the final solution.