Solve the differential equation y''-2y'+y= 0
Answers
Answer :
Find all solutions of the second-order linear differential equation
\[ y'' + 2y' + y = 0 \]
on the interval (-\infty, +\infty).
The given second-order linear differential equation is of the form
\[ y'' + ay' + by = 0 \qquad \text{with} \qquad a = 2, \quad b = 1.\]
These values of a and b give us d = a^2 - 4b = 0. By Theorem 8.7 (page 326-327 of Apostol) we then have
\[ y = e^{-\frac{ax}{2}} \left( c_1 u_1(x) + c_2 u_2(x) \right) \qquad \text{where} \qquad u_1 (x) = 1, \quad u_2 (x) = x. \]
Therefore,
\begin{align*} y &= e^{-x} \left( c_1 + c_2 x \right) \end{align*}
for constants c_1 and c_2.
y” + 2y’ + y = 0
(D2 + 2D +1) y = 0
⇒ D2 + 2D + 1 = 0
ie, D = -1, -1
∴ y = (C1 + C2x)e-x
Applying boundary conditions, we get
⇒ 0 = [C1 + C2(0)]e-0
⇒ C1 = 0
0 = (C1 + C2)e-1
⇒ C1 + C2 = 0
⇒ C2 = 0