Definitions
ODE (Ordinary Differential Equation): An equation involving a function of one variable and its derivatives.
IVP (Initial Value Problem): An ODE together with conditions such as \( y(t_0)=y_0 \), \( y^{\prime}(t_0)=y_1 \).
Second-Order Linear Equation
\[
a_2(t)y^{\prime\prime} + a_1(t)y^{\prime} + a_0(t)y = g(t)
\]
\[
y^{\prime\prime} + p(t)y^{\prime} + q(t)y = g(t)
\]
Homogeneous case:
\[
y^{\prime\prime} + p(t)y^{\prime} + q(t)y = 0
\]
Constant Coefficients
\[
y^{\prime\prime} + ay^{\prime} + by = 0
\]
\[
r^2 + ar + b = 0
\]
Distinct roots: \( y = C_1 e^{r_1 t} + C_2 e^{r_2 t} \)
Repeated root: \( y = (C_1 + C_2 t)e^{rt} \)
Complex roots \( r=\alpha \pm \beta i \):
\[
y = e^{\alpha t}(C_1 \cos(\beta t) + C_2 \sin(\beta t))
\]
Wronskian
\[
W[y_1,y_2](t) =
\begin{vmatrix}
y_1 & y_2 \\
y_1^{\prime} & y_2^{\prime}
\end{vmatrix}
=
y_1 y_2^{\prime} - y_2 y_1^{\prime}
\]
If \( W(t_0)\neq0 \), then \( y_1,y_2 \) are linearly independent.
General solution from fundamental set:
\[
y = C_1 y_1 + C_2 y_2
\]
Abel’s Formula
\[
W(t) = C e^{-\int p(t)\,dt}
\]
Existence and Uniqueness Theorem
\[
y^{\prime\prime} + p(t)y^{\prime} + q(t)y = g(t), \quad
y(t_0)=y_0, \quad y^{\prime}(t_0)=y_1
\]
If \( p(t), q(t), g(t) \) are continuous on an open interval containing \( t_0 \), then there exists a unique solution on that interval.
Reduction of Order
\[
y_2 = y_1 \int \frac{e^{-\int p(t)\,dt}}{y_1^2}\,dt
\]
Euler–Cauchy Equation
\[
t^2 y^{\prime\prime} + at y^{\prime} + by = 0
\]
\[
r(r-1) + ar + b = 0
\]
Distinct roots: \( y=C_1 t^{r_1}+C_2 t^{r_2} \)
Repeated root: \( y=C_1 t^r + C_2 t^r \ln t \)
Complex roots \( r=\alpha \pm \beta i \):
\[
y = t^{\alpha}\left(C_1 \cos(\beta \ln t) + C_2 \sin(\beta \ln t)\right)
\]