Lecture4.1.pdf
ENGR 232
Dynamic Engineering
Systems
Lecture 4.1
Dr. Michael Ryan
Agenda
• Quick Review
– First Order Differential Equations
• Models
• Second Order Differential Equations
– Models
– Homogeneous equations
– Auxiliary equation and its roots
– Unique solutions
21/20/2016 ENGR 232 Winter 2016 Lecture 4.1 – Dr. M. Ryan
2nd and higher order Linear D.E
Based on our initial knowledge of linear and nonlinear 1st order differential equations
• Investigate Linear 2nd Order D.E. and
Solution
s
– Linear Differential Equations
– Phase plane (similar to dfield8 but 2D – go to Rice website)
𝑥′𝑣𝑠.𝑥 𝑜𝑟 𝑥2 𝑣𝑠 𝑥1
– Laplace Transform (notes and Workshop)
– Multivariate Systems (notes matrix formulation)
– And lots more
1/20/2016 ENGR 232 Winter 2016 Lecture 4.1 – Dr. M. Ryan Page 3
2nd Order Homogeneous Linear D.E.
With Constant Coefficients
Standard equation:
Homogeneous equation:
This equation always has a solution of the type y(t) = ert.
Define Characteristic (Auxiliary) equation:
Roots: Discriminant:
If D > 0, If D < 0, If D = 0
ENGR 232 Winter 2016 Lecture 4.1 – Dr. M. Ryan Page 4
a ¢¢y + b ¢y + cy = f (t), a ¹ 0
D = b
2
- 4ac
1/20/2016
f(t) is called the forcing function
f(t) is set to 0
can get by inspection of
D.E Let r = d/dt
2 real distinct roots 2 complex conjugate roots 2 repeated real roots
𝒂𝒚′′ + 𝒃𝒚′ + 𝒄𝒚 = 𝟎
𝒂𝒚′′ + 𝒃𝒚′ + 𝒄𝒚 = 𝟎 = 𝐚𝐫𝟐𝐞𝐫𝐭 + 𝐛𝐫𝐞𝐫𝐭 + 𝐜𝐞𝐫𝐭 → 𝒂𝒓𝟐 + 𝒃𝒓 + 𝒄 𝒆𝒓𝒕 = 𝟎
𝒂𝒓𝟐 + 𝒃𝒓 + 𝒄 = 𝟎
𝒓 =
−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄
𝟐𝒂
Two Independent
...
1. Lecture4.1.pdf
ENGR 232
Dynamic Engineering
Systems
Lecture 4.1
Dr. Michael Ryan
Agenda
• Quick Review
– First Order Differential Equations
• Models
• Second Order Differential Equations
– Models
– Homogeneous equations
– Auxiliary equation and its roots
– Unique solutions
21/20/2016 ENGR 232 Winter 2016 Lecture 4.1 – Dr. M. Ryan
2. 2nd and higher order Linear D.E
Based on our initial knowledge of linear and nonlinear 1st order
differential equations
• Investigate Linear 2nd Order D.E. and
Solution
s
– Linear Differential Equations
– Phase plane (similar to dfield8 but 2D – go to Rice website)
�′��.� �� �2 �� �1
– Laplace Transform (notes and Workshop)
– Multivariate Systems (notes matrix formulation)
– And lots more
1/20/2016 ENGR 232 Winter 2016 Lecture 4.1 – Dr. M. Ryan
Page 3
3. 2nd Order Homogeneous Linear D.E.
With Constant Coefficients
Standard equation:
Homogeneous equation:
This equation always has a solution of the type y(t) = ert.
Define Characteristic (Auxiliary) equation:
Roots: Discriminant:
If D > 0, If D < 0, If D = 0
ENGR 232 Winter 2016 Lecture 4.1 – Dr. M. Ryan Page 4
a ¢¢y + b ¢y + cy = f (t), a ¹ 0
D = b
2
- 4ac
4. 1/20/2016
f(t) is called the forcing function
f(t) is set to 0
can get by inspection of
D.E Let r = d/dt
2 real distinct roots 2 complex conjugate
roots 2 repeated real roots
��′′ + ��′ + �� = �
��′′ + ��′ + �� = � = ������ + ����� + ���� →
��� + �� + � ��� = �
��� + �� + � = �
� =
−� ± �� − ���
��
