Linear Ordinary Differential Equations

Introduction Linear ordinary differential equations are ordinary differential equations (differential equations with only one independent variable) of the form an (x)y (n) (x) + · · · + a1 (x)y(x) + a0 (x)y(x) = b(x)

Homogeneous Linear ODE’s A homogeneous linear ODE is a linear ODE in which b(x) = 0. The only general method for solving such equations is via series.

The Principle of Superposition and General Solutions Due to the magic of linear algebra, it can be shown that the sum of individual solutions to a linear ODE is also a solution to that linear ODE. This is called the Principle of Superposition (PoS) and is extremely important. In fact because of this theorem, it has been proven that the general solution to a linear ODE is the sum of the homogeneous and particular solutions. Algebraically, yg = yh + yp

Example Solution Find the general solution to y 00 − 2y 0 − 8y = sin x + cos x

Homogeneous Solution Let’s guess that yh is in the form yh = eλx – this seems like a good guess, doesn’t it? So, plugging in, we have
λ2 − 2λ − 8 eλx = 0 Now, since eλx is never 0, we can solve the above equation as a quadradic, and we find that either λ = 4 or λ = −2. Therefore, by the PoS, yh = c1 e4x + c2 e−2x where c1 and c2 are arbitrary constants.

Particular Solution by the Method of Undetermined Coefficients Let’s guess that yp is in the form yp = A sin x + B cos x. The object of the method of undetermined coefficients is to solve for A and B. Because of the 1

PoS, we are able to solve for both A and B by plugging in their respective terms for y. So, solving for A, we have −A sin x − 2A cos x − 8A sin x = sin x + cos x which gives A=−

sin x + cos x 9 sin x + 2 cos x

and solving for B, we have −B cos x + 2B sin x − 8B cos x = sin x + cos x which gives B=−

sin x + cos x 9 cos x − 2 sin x

General Solution Finally, combining the homogeneous and particular solutions, we have y = c1 e4x + c2 e−2x −

sin x + cos x sin x + cos x − 9 sin x + 2 cos x 9 cos x − 2 sin x

More on the Method of Undetermined Coefficients Although the method of undetermined coefficients is comparatively easy, it only works for a very small class of linear ODE’s with constant coefficients. For this approach to be successful, c(x) must be in the form X c(x) = Ai pi (x)eai x sin bi x cos ci x i

where {pi (x)} are polynomials and {Ai } are the coefficients to be solved for.

The Method of Variation of Parameters Although the method of variation of parameters is the most general method for determining particular solutions, it is terribly annoying to derive, so I won’t do so. Anyways, the gist of this method (for second-order equations) is to solve the system of equations u01 y1

+u02 y2

=0

u01 y10

+u02 y20

= c(x)

for u01 and u02 then integrate and plug into yp = u 1 y1 + u 2 y2

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