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Last updated 9/4/2017 ODE Cheat Sheet
ODE Cheat Sheet for Math 331
Scalar linear first-order ODEs
Consider the ODE, x ˙ = a ( t ) x + f ( t ).
Setting
A ( t ) =
a ( t ) d t,
the general solution is given by the variation of parameters formula,
x ( t ) = c 1 e A ( t )^ + e A ( t )
e− A ( t ) f ( t ) d t.
In the special case that a ( t ) ≡ a , the solution becomes
x ( t ) = c 1 e at^ + e at
e− atf ( t ) d t.
Scalar linear second-order homogeneous ODEs
Consider the homogeneous ODE, x ¨ + p x ˙ + qx = 0_._
The characteristic equation is λ^2 + pλ + q = 0_._
Let λ 1 , λ 2 be the roots of the characteristic equation. If λ 1 , λ 2 are real, the homogeneous solution is
x h( t ) = c 1 e λ^1 t^ + c 2 e λ^2 t.
If λ 1 = λ 2 = − p/ 2, which requires p^2 = 4 q , the homogeneous solution is
x h( t ) = c 1 e− pt/^2 + c 2 t e− pt/^2_._
If λ 1 = a + i b with b , 0, the homogeneous solution is
x h( t ) = c 1 e at^ cos( bt ) + c 2 e at^ sin( bt ).
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ODE Cheat Sheet
Scalar linear second-order nonhomogeneous ODEs
Consider the nonhomogeneous ODE, x ¨ + p x ˙ + qx = f ( t ).
The solution is the sum of the homogeneous and particular solutions,
x ( t ) = x h( t ) + x p( t ).
Write the homogeneous solution as x h( t ) = c 1 x 1 ( t ) + c 2 x 2 ( t ) ,
and set Φ( t ) = x 1 ( t ) x ′ 2 ( t ) − x 2 ( t ) x ′ 1 ( t ).
The particular solution is given by the variation of parameters formula,
x p( t ) = −
x 2 ( t ) f ( t ) Φ( t ) d t
x 1 ( t ) +
x 1 ( t ) f ( t ) Φ( t ) d t
x 2 ( t ).
First-order homogeneous linear system of ODEs
Consider the first-order system,
x ˙ = Ax, A =
a b c d
The eigenvalues of A are λ 1 , λ 2 , and the associated eigenvectors are v 1 , v 2. If the eigenvalues are real the general solution is, x ( t ) = c 1 e λ^1 tv 1 + c 2 e λ^2 tv 2_._
If λ 1 = a + i b ( b , 0) with associated eigenvector v 1 = p + i q , the general solution is,
x ( t ) = c 1 e at^ (cos( bt ) p − sin( bt ) q ) + c 2 e at^ (sin( bt ) p + cos( bt ) q ).
We have the following classification of the fixed point, x = 0 :
(a) λ 1 < 0 < λ 2 : unstable saddle point
(b) λ 1 < λ 2 < 0: stable node
(c) 0 < λ 1 < λ 2 : unstable node
(d) if λ 1 = a + i b ( b , 0),
- a < 0: stable spiral
- a > 0: unstable spiral
- a = 0: linear center.
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