Lecture Notes in Differential Equations - Bruce E. Shapiro

413
The resulting equation is linear and
y(t) =
[ ( ∫ 1
C +
µ
)] 1/(1−n)
µ(t)(1 − n)q(t)dt
where
( ∫
µ(t) = exp (1 − n)
)
p(t)dt
Second Order Homogeneous Linear Equation with Constant
Coefficients
To solve the differential equation
ay ′′ + by ′ + cy = 0
find the roots of the characteristic equation
ar 2 + br + c = 0
If the roots (real or complex) are distinct, then
If the roots are repeated then
y = Ae r1t + Be r2t
y = (A + Bt)e rt
Method of Undetermined Coefficients
To solve the differential equation
ay ′′ + by ′ + cy = f(t)
where f(t) is a polynomial, exponential, or trigonometric function, or any
product thereof, the solution is
y = y H + y P
where y H is the complete solution of the homogeneous equation
ay ′′ + by ′ + cy = 0