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Underlying Theorem Example (Equation) Example (System)
Translating Higher OrderEquations/Systems to First Order Systems
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
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Underlying Theorem Example (Equation) Example (System)
Intermediate Derivatives Become DummyVariables
Let a0(t), . . . ,an−1(t) and g(t) be differentiable functions. Thefunction y is a solution of the nth order linear differentialequation y(n) +an−1(t)y(n−1) + · · ·+a1(t)y′ +a0(t)y = g(t) ifand only if y0 := y, y1 := y′, . . . , yn−1 := y(n−1) is a solution ofthe system of linear equations
y′0 = y1
y′1 = y2...
...y′n−2 = yn−1
y′n−1 = −an−1(t)yn−1 −·· ·−a1(t)y1 −a0(t)y0 +g(t).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Intermediate Derivatives Become DummyVariables
Let a0(t), . . . ,an−1(t) and g(t) be differentiable functions. Thefunction y is a solution of the nth order linear differentialequation y(n) +an−1(t)y(n−1) + · · ·+a1(t)y′ +a0(t)y = g(t) ifand only if y0 := y, y1 := y′, . . . , yn−1 := y(n−1) is a solution ofthe system of linear equations
y′0 = y1
y′1 = y2...
...y′n−2 = yn−1
y′n−1 = −an−1(t)yn−1 −·· ·−a1(t)y1 −a0(t)y0 +g(t).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
Idea: y0 = y, y1 = y′
y1 = y′0y′1 +4y1 +5y0 = 0
y0(0) = 0y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
Idea: y0 = y, y1 = y′
y1 = y′0y′1 +4y1 +5y0 = 0
y0(0) = 0y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
Idea: y0 = y, y1 = y′
y1 = y′0
y′1 +4y1 +5y0 = 0y0(0) = 0y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
Idea: y0 = y, y1 = y′
y1 = y′0y′1
+4y1 +5y0 = 0y0(0) = 0y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
Idea: y0 = y, y1 = y′
y1 = y′0y′1 +4y1
+5y0 = 0y0(0) = 0y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
Idea: y0 = y, y1 = y′
y1 = y′0y′1 +4y1 +5y0
= 0y0(0) = 0y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
Idea: y0 = y, y1 = y′
y1 = y′0y′1 +4y1 +5y0 = 0
y0(0) = 0y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
Idea: y0 = y, y1 = y′
y1 = y′0y′1 +4y1 +5y0 = 0
y0(0) = 0
y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
Idea: y0 = y, y1 = y′
y1 = y′0y′1 +4y1 +5y0 = 0
y0(0) = 0y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
y′0 = y1
y′1 = −4y1 −5y0
y0(0) = 0y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
y′0 = y1
y′1 = −4y1 −5y0
y0(0) = 0y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
y′0 = y1
y′1 = −4y1 −5y0
y0(0) = 0y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
y′0 = y1
y′1 = −4y1 −5y0
y0(0) = 0
y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′ +4y′ +5y = 0, y(0) = 0, y′(0) = 1 Into an IVPfor a First Order System
y′0 = y1
y′1 = −4y1 −5y0
y0(0) = 0y1(0) = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2
u2 = u′1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2
+3u2 −u1 +2u4 = 0u4 = u′
3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2
−u1 +2u4 = 0u4 = u′
3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1
+2u4 = 0u4 = u′
3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4
= 0u4 = u′
3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4
−2u3 +u2 −2u1 = 0u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4 −2u3
+u2 −2u1 = 0u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4 −2u3 +u2
−2u1 = 0u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4 −2u3 +u2 −2u1
= 0u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0
u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0 u2(0) = 1
u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0 u2(0) = 1u3(0) = 2
u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
Idea: u1 := y1, u2 := y′1, u3 := y2, u4 := y′2u2 = u′
1
u′2 +3u2 −u1 +2u4 = 0
u4 = u′3
u′4 −2u3 +u2 −2u1 = 0
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
u′1 = u2
u′2 = −3u2 +u1 −2u4
u′3 = u4
u′4 = 2u3 −u2 +2u1
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
u′1 = u2
u′2 = −3u2 +u1 −2u4
u′3 = u4
u′4 = 2u3 −u2 +2u1
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
u′1 = u2
u′2 = −3u2 +u1 −2u4
u′3 = u4
u′4 = 2u3 −u2 +2u1
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
u′1 = u2
u′2 = −3u2 +u1 −2u4
u′3 = u4
u′4 = 2u3 −u2 +2u1
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
u′1 = u2
u′2 = −3u2 +u1 −2u4
u′3 = u4
u′4 = 2u3 −u2 +2u1
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
u′1 = u2
u′2 = −3u2 +u1 −2u4
u′3 = u4
u′4 = 2u3 −u2 +2u1
u1(0) = 0
u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
u′1 = u2
u′2 = −3u2 +u1 −2u4
u′3 = u4
u′4 = 2u3 −u2 +2u1
u1(0) = 0 u2(0) = 1
u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
u′1 = u2
u′2 = −3u2 +u1 −2u4
u′3 = u4
u′4 = 2u3 −u2 +2u1
u1(0) = 0 u2(0) = 1u3(0) = 2
u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
logo1
Underlying Theorem Example (Equation) Example (System)
Translate the Initial Value Problemy′′1 +3y′1 − y1 +2y′2 = 0, y′′2 −2y2 + y′1 −2y1 = 0,y1(0) = 0, y′1(0) = 1, y2(0) = 2, y′2(0) = 3 Into anIVP for a First Order System
u′1 = u2
u′2 = −3u2 +u1 −2u4
u′3 = u4
u′4 = 2u3 −u2 +2u1
u1(0) = 0 u2(0) = 1u3(0) = 2 u4(0) = 3
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Translating Higher Order Equations/Systems to First Order Systems
