Chapter 38 DIFFERENTIAL EQUATIONS 952021 differential equations by
Chapter 38 DIFFERENTIAL EQUATIONS 9/5/2021 differential equations by Chtan (FYHS-Kulai) 1
What is a differential equation ? (D. E. ) In short for 2 differential equations by Chtan (FYHS-Kulai) 9/5/2021
A differential equation is an equation that involves one or more derivatives, or differentials. 3 differential equations by Chtan (FYHS-Kulai) 9/5/2021
4 Differential equations play a prominent role in engineering, physics, economics, and other disciplines. differential equations by Chtan (FYHS-Kulai) 9/5/2021
Differential equations are classified by 3 components : (a) type : ordinary or partial (b) order : the order of the highest-order derivative that occurs in the equation 5 differential equations by Chtan (FYHS-Kulai) 9/5/2021
(c) degree : the exponent of the highest power of the highest-order derivative, after the equation has been cleared of fractions and radicals in the dependent variable and its derivatives. 6 differential equations by Chtan (FYHS-Kulai) 9/5/2021
For example, is an ordinary differential equation, of order 3 and degree 2. 7 differential equations by Chtan (FYHS-Kulai) 9/5/2021
If the dependent variable y is a function of a single independent variable x, say y=f(x) 8 Only “ordinary” derivatives occur ! differential equations by Chtan (FYHS-Kulai) 9/5/2021
If the dependent variable y is a function of 2 or more independent variables, say if x and t are independent variables, then partial derivatives of y may occur. 9 differential equations by Chtan (FYHS-Kulai) 9/5/2021
For example, is a partial differential equation, of order 2 and degree one. [this is the 1 -dimensional “wave-equatio 10 differential equations by Chtan (FYHS-Kulai) 9/5/2021
For a discussion of partial differential equations, including the wave equation and solutions of associated physical problems, see Kaplan, Advanced Calculus, Chapter 10. 11 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Before we proceed to solve D. E. , let us first examine how to form a D. E. from an ordinary equation including normal functions and trigonometric function or exponential function. 12 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Let us see the following equation, Then, This is an D. E. 13 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 1 Eliminate the arbitrary constant A from the equation : 14 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : Integrating both sides w. r. t. x 15 differential equations by Chtan (FYHS-Kulai) 9/5/2021
16 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 2 Eliminate the arbitrary constants A and B from the equation : 17 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : 18 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Do p 250 and p 251 Ex 19 a 19 differential equations by Chtan (FYHS-Kulai) 9/5/2021
20 We will focus on ordinary differential equations(no partial D. E. ). In outline, these are things we will study : differential equations by Chtan (FYHS-Kulai) 9/5/2021
1. First-order equations (a) variables separable (b) homogeneous (c) linear 2. Special types of second-order equations 21 differential equations by Chtan (FYHS-Kulai) 9/5/2021
nd 3. 2 22 order Linear equations with constant coefficients (a) homogeneous (b) differential equations by Chtan (FYHS-Kulai) 9/5/2021
A First-order equations with variables separable 23 differential equations by Chtan (FYHS-Kulai) 9/5/2021
A first-order differential equation can be solved by integration if it is possible to collect all y terms with dy and all x terms with dx. It is possible to write the equation in the form 24 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Then the general solution (G. S. ) is where C is an arbitrary constant. 25 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 3 Solve the equation 26 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : 27 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Do p. 253 Ex 19 28 differential equations by Chtan (FYHS-Kulai) 9/5/2021
B First-order homogeneous equations 29 differential equations by Chtan (FYHS-Kulai) 9/5/2021
A differential equation that can be put into the form is said to be homogeneous. Such an equation can be solved by introducing a new dependent variable 30 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Then , Solved by separation of variables : 31 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 4 Show that the equation is homogeneous, and solve it. 32 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : From the given equation, we have 33 differential equations by Chtan (FYHS-Kulai) 9/5/2021
The solution of this is : 34 differential equations by Chtan (FYHS-Kulai) 9/5/2021
35 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Do p. 255 Ex 19 36 differential equations by Chtan (FYHS-Kulai) 9/5/2021
C First-order linear equations 37 differential equations by Chtan (FYHS-Kulai) 9/5/2021
is of the first degree is of the second degree If every term of a differential equation is of degree zero or degree one, then the equation is linear. 38 differential equations by Chtan (FYHS-Kulai) 9/5/2021
A linear differential equation of first order can always be put into the standard form : where P and Q are functions of x. 39 differential equations by Chtan (FYHS-Kulai) 9/5/2021
st 1 * order Liner equations are solved by multiplying throughout by the function : is known as an integrating factor. 40 differential equations by Chtan (FYHS-Kulai) 9/5/2021
41 differential equations by Chtan (FYHS-Kulai) 9/5/2021
42 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 5 Solve the equation 43 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : This is of the form with P=-1, Q=x Now, multiplying both sides b 44 differential equations by Chtan (FYHS-Kulai) 9/5/2021
45 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Integration by parts Multiply both sides by 46 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 6 If and Find y in terms of x. 47 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : The equation is of the form 48 differential equations by Chtan (FYHS-Kulai) 9/5/2021
The G. S. is : 49 differential equations by Chtan (FYHS-Kulai) 9/5/2021
50 differential equations by Chtan (FYHS-Kulai) 9/5/2021
51 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Do p. 258 Ex 19 52 differential equations by Chtan (FYHS-Kulai) 9/5/2021
D 53 Special types of secondorder equations differential equations by Chtan (FYHS-Kulai) 9/5/2021
2 types of equation will be considered : 54 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Equations of the form : These equations are solvable by integrating twice w. r. t. x 55 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 7 Solve the equation Given that B, W and i are constants. When x=0, y=0 and dy/dx=0. 56 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : Integrating both sides w. r. t. x 57 differential equations by Chtan (FYHS-Kulai) 9/5/2021
When y=0, x=0, dy/dx=0, Hence, c=0. Again, Integrating both sides w. r. t. x 58 differential equations by Chtan (FYHS-Kulai) 9/5/2021
When y=0, x=0, dy/dx=0, Hence, c 1=0. 59 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 8 Solve the equation 60 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 9 Solve the equation 61 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Equations of the form : 62 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Firstly, multiplying both sides of the equation by 63 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Next, integrating both sides of the equation w. r. t. x 64 differential equations by Chtan (FYHS-Kulai) 9/5/2021
After the integration of this equation, the second integration is needed by separating the variables. 65 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 10 Solve the equation 66 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : Then, integrating both sides w. r. t 67 differential equations by Chtan (FYHS-Kulai) 9/5/2021
68 differential equations by Chtan (FYHS-Kulai) 9/5/2021
69 differential equations by Chtan (FYHS-Kulai) 9/5/2021
70 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 11 Solve the equation 71 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Do p. 260 Ex 19 72 differential equations by Chtan (FYHS-Kulai) 9/5/2021
E 73 Homogeneous nd linear 2 order differential equations with constant coefficients differential equations by Chtan (FYHS-Kulai) 9/5/2021
Where a, b, c are constants Coefficients. 74 differential equations by Chtan (FYHS-Kulai) 9/5/2021
This equation has a solution if 75 This is called the characteristic equation of the D. E. Some books called this auxiliary equation. differential equations by Chtan (FYHS-Kulai) 9/5/2021
This equation has 3 cases to be considered. 1. Roots are real and different 2. Roots are real and equal 3. Roots are imaginary 76 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Case 1 Let the roots be Then are solutions of the D. E. constants and the G. S. is 77 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 12 Find the general solution (G. S. ) of the equation 78 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : Characteristic equation or auxiliary equation is : 79 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Case 2 Let the roots be Then the G. S. is where A and B are arbitrary constants. 80 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 13 Solve the equation 81 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : Characteristic equation or auxiliary equation is : 82 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Case 3 Let the roots be Where 83 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 14 Solve the equation 84 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : 85 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Do p. 263 Ex 19 86 differential equations by Chtan (FYHS-Kulai) 9/5/2021
F Nonhomogeneous 87 linear second-order differential equations with constant coefficients differential equations by Chtan (FYHS-Kulai) 9/5/2021
Where a, b, c are constants 88 differential equations by Chtan (FYHS-Kulai) 9/5/2021
The general solution of the equations is made up of the sum of 2 parts. The G. S. of the homogeneous equation 89 The particular integral differential equations by Chtan (FYHS-Kulai) 9/5/2021
This part is also called the Complementary Function To find this part of the solution : we 90 use the methods learnt in previous section. By letting f(x)=0. differential equations by Chtan (FYHS-Kulai) 9/5/2021
We consider 4 cases to find this Particular Integral. 91 Case 1 : f(x)=a polynoimial of degree n. Case 2 : Case 3 : Case 4 : f(x)=the mixture of the above cases differential equations by Chtan (FYHS-Kulai) 9/5/2021
Case 1 : A particular integral can be found by substituting 92 The constants can be determined by equating coefficients. differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 15 Find a particular integral of the equation and hence write down the general solution. 93 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln Let : 94 differential equations by Chtan (FYHS-Kulai) 9/5/2021
The complementary function is the GS of the equation 95 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Hence the GS of the given equation is : 96 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Case 2 : A particular integral can be found by substituting The constant p can be determined by equating coefficients. 97 differential equations by Chtan (FYHS-Kulai) 9/5/2021
If the function occurs in the complementary function, p will be indeterminate. We have to try or even have to try. 98 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 16 Find the GS of the equations : 99 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : (i) The complementary function is the GS of the equation 100 differential equations by Chtan (FYHS-Kulai) 9/5/2021
i. e. Let a particular integral be 101 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Hence, a particular integral is The GS is Complementary Fn 102 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : (ii) The complementary function is the GS of the equation We must let a particular integr 103 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Sub into the original eqn. 104 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Hence, a particular integral is The G. S. is 105 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Case 3 : A particular integral can be found by substituting The constants p and q can be determined by equating coefficients. 106 differential equations by Chtan (FYHS-Kulai) 9/5/2021
If the functions occur in the complementary function. We have to try the following functions : 107 differential equations by Chtan (FYHS-Kulai) 9/5/2021
e. g. 17 Solve the equation 108 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : To find the complementary function : 109 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Sub into the original eqn. 110 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Equating coefficients of cos 2 x and sin 2 x, A paticular integral is The G. S. is 111 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Case 4 : A combination of the previous types. e. g. 18 Obtain the solution of the equation for which y=0, dy/dx=1 when x=0. 112 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Soln : The complementary function is 113 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Let a particular integral be 114 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Sub into the original eqn. 115 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Equating coefficients, 116 differential equations by Chtan (FYHS-Kulai) 9/5/2021
A paticular integral is The G. S. is 117 differential equations by Chtan (FYHS-Kulai) 9/5/2021
When x=0, y=0, dy/dx=1, 118 differential equations by Chtan (FYHS-Kulai) 9/5/2021
The required solution is 119 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Do p. 267 Ex 19 120 differential equations by Chtan (FYHS-Kulai) 9/5/2021
Discard these : 19 d (32 -35) 121 Do these : 19 e (1, 3, 5, 7, 9, 11, 13, 15) 19 f (odd # 1 -19) Misc. Ex. (odd # 5 -25) differential equations by Chtan (FYHS-Kulai) 9/5/2021
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