104
Differential Equations
Prepared by: Midori Kobayashi Humber College
35
1Click on the Computer Image at the bottom right for a direct
web link to Wikipedia Math.35.1 Definitions
35.1 - EXAMPLE 4 Page 1031
the order of the highest - order derivative in the equation
First derivativeSecond derivativeFirst derivative
Third derivativeSecond derivative
35.1 - EXAMPLE 5 Page 1031
The degree of a differential equation is the degree of the
highest - order derivative in the equation.
*degree : the power to which that derivative is raised
2nd power35.1 - EXAMPLE 6 Page 1031
1st poweronly one derivative
the degree of the highest - order derivative in the equation
Clear the fraction and square both sides
the order of the highest - order derivative in the equation
Highest - order derivative
35.2 Graphical and Numerical Solution of Differential
Equations
35.2 - EXAMPLE 9 Page 1034
Replacing dy/dx with m
(cont)35.2 - EXAMPLE 9 Page 1034 - Continued
At (5,0) the slope is 5At (3,2) the slope is 1
(cont)35.2 - EXAMPLE 9 Page 1034 - Continued
35.2 - EXAMPLE 10 Page 1035
35.3 First-Order Differential Equation, Variables Separable35.3
- EXAMPLE 12 Page 1038
(cont)
35.3 - EXAMPLE 12 Page 1038 - Continued
35.3 - EXAMPLE 13 - Page 1039
(cont)
35.3 - EXAMPLE 13 - Page 1039 - Continued
Change from Log. form to Exp. form
35.3 - EXAMPLE 14 - Page 1039
(cont)35.3 - EXAMPLE 14 - Page 1039 - Continued
Log property: LogA + LogB = LogAB
u = 5 x du = dx
Change from Log. form to Exp. form35.4 Exact First-Order
Differential Equation
35.4 - EXAMPLE 17 - Page 1042
The variables are not separable Product rule
35.4 - EXAMPLE 18 - Page 1042
(cont)
35.4 - EXAMPLE 18 - Page 1042 - continued
35.4 - EXAMPLE 19 - Page 1043
35.5 First - Order Homogeneous Differential Equations
35.5 - EXAMPLE 22 - Page 1044
every term is of the same degree
2nd degree
All second - degree
1st degree 35.5 - EXAMPLE 23 - Page 1045
Every term is of the same degree
Multiply each term by dx
MN(cont)35.5 - EXAMPLE 23 - Page 1045 - Continued
MN
Of first degree
Of first degree(cont)35.5 - EXAMPLE 23 - Page 1045 -
Continued
Product Rule!
(cont)35.5 - EXAMPLE 23 - Page 1045 - Continued
(cont)35.5 - EXAMPLE 23 - Page 1045 - Continued
u = v, a =1, and du = dv
(cont)
35.5 - EXAMPLE 23 - Page 1045 - Continued
35.6 First-Order Linear Differential Equations
35.6 - EXAMPLE 25 - Page 1047
dy/dx P Q35.6 - EXAMPLE 26 - Page 1047
By log property logAp =PlogA
(cont)35.6 - EXAMPLE 26 - Page 1047 - Continued
By log property
Multiply by dx
(cont)
35.6 - EXAMPLE 26 - Page 1047 - Continued
35.7 Geometric Applications of First - Order Differential
Equations
35.7 - EXAMPLE 31 Page 1052
10C1(cont)35.7 - EXAMPLE 31 Page 1052 - Continued
Change from Log form to Exp. form
35.7 - EXAMPLE 32 Page 1052
(cont)35.7 - EXAMPLE 32 Page 1052 - Continued
35.8 Exponential Growth and Decay
35.8 - EXAMPLE 34 Page 1055
a constant of proportionality
u = a y du = dy
(cont)35.8 - EXAMPLE 34 Page 1055 - Continued
1
35.8 - EXAMPLE 35 Page 1056
a constant of proportionality
(a) W: weight of the crate g: 9.806 m/s2dv/dt: the
acceleration
(cont)35.8 - EXAMPLE 35 Page 1056 - Continued
|v| = 66.4
Multiply by 66.4/W
(cont) W: weight of the crate g: 9.806 m/s2dv/dt: the
acceleration
35.8 - EXAMPLE 35 Page 1056 - Continued
u = 66.4 - vdu = dv
(cont)35.8 - EXAMPLE 35 Page 1056 - Continued
35.9 Series RL and RC Circuits
35.9 - EXAMPLE 37 Page 1058
(cont)35.9 - EXAMPLE 37 Page 1058 - Continued
(cont)35.9 - EXAMPLE 37 Page 1058 - Continued
(cont)35.9 - EXAMPLE 37 Page 1058 - Continued
35.9 - EXAMPLE 38 Page 1059
(cont)
35.9 - EXAMPLE 38 Page 1059 - Continued
35.10 Second-Order Differential Equations
35.10 - EXAMPLE 40 Page 1062
(cont)35.10 - EXAMPLE 40 Page 1062 - Continued
35.10 - EXAMPLE+
(cont)35.10 - EXAMPLE+ - Continued
35.11 Second-Order Differential Equations with Constant
Coefficients and Right Side Zero
35.11 - EXAMPLE 41 Page 1064
a = 1, b = 3, and c = 2
35.11 - EXAMPLE 42 Page 1064
a = 1, b = 5, and c = 0
35.11 - EXAMPLE 44 Page 1065
a = 1, b = 5, and c = 0
35.12 Second-Order Differential Equations with right Side Not
Zero
35.12 - EXAMPLE 49 Page 1071
Try a solution consisting of the sum of f (x), f (x), and f(x),
each with an (as yet) undetermined (constant) coefficient.
the particular integral will make the left side equal to 3x
(cont)35.12 - EXAMPLE 49 Page 1071 - Continued
35.12 - EXAMPLE 49+
35.12 - EXAMPLE 51 Page 1072
(cont)
35.12 - EXAMPLE 51 Page 1072 - Continued
Try a solution consisting of the sum of f (x), f (x), and f(x),
each with an (as yet) undetermined (constant) coefficient.
35.12 - EXAMPLE 51 Page 1072 - Continued
7(cont)
35.12 - EXAMPLE 51 Page 1072 - Continued
35.13 RLC Circuits
35.13 - EXAMPLE 53 Page 1076
