Lecture Notes in Differential Equations - Bruce E. Shapiro

140 LESSON 16. LINEAR EQS. W/ CONST. COEFFICENTS
Proof. We are give Ly H = 0 and Ly P = f(t). Hence
Ly = L(y H + y P ) = Ly H + Ly P = 0 + f(t) = f(t) (16.52)
Hence y = h H + y P is a solution.
General Principal. The general solution to
Ly = ay ′′ + by ′ + cy = f(t) (16.53)
is the sum of a homogeneous and a particular part:
where Ly H = 0 and Ly P = f(t).
y = y H (t) + y P (t) (16.54)
Theorem 16.8. Principle of Superposition If y H1 (t) and y H2 (t) are
both solutions of Ly = 0, then any linear combination
is also a solution ofLy = 0.
Proof. Since y H1 (t) and y H2 (t) are solutions,
Since L is a linear operator,
y H (t) = Ay H1 (t) + By H2 (t) (16.55)
Ly H1 = 0 = Ly H2 (16.56)
Ly H = L[Ay H1 + By H2 ] (16.57)
= ALy H1 + BLy H2 (16.58)
= 0 (16.59)
Hence any linear combination of solutions to the homogeneous equation is
also a solution of the homogeneous equation.
General Solution of the Homogeneous Equation with Constant
Coefficients. From theorem (16.4) we know that e rt is a solution of Ly = 0
whenever r is a root of the characteristic equation. If r is a repeated root,
we also know from theorem (16.5) that te rt is also a solution. Thus we
can always find two solutions to the homogeneous equation with constant
coefficients by finding the roots of the characteristic equation. In general
these are sufficient to specify the complete solution.