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ODEs
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A quick review…
First Order ODE’sLinear ODE is of the form
Solution can be derived using integrating factor method i ti f t th dor variation of parameter method
Separable Equation:
First Order ODE’s (…cont)Homogeneous Equation*:
Results in a separable equation
*Homogeneous equation can also mean a linear ODE that has zero as its “forcing function” on the right‐hand side
First Order ODE’s (…cont)Bernouilli Equation:
By making the substitution
lWe can get a linear ODE:
First Order ODE’s (…cont)Exact Equation:
Thus we seek a solutionTest for exactness:Test for exactness:
Solve for F(x,y):( y)
Now we need to solve for A(y)Now we need to solve for A(y)
First Order ODE’s (…cont)
Thus, the implicit solution is:
First Order ODE’s (…cont)Integrating Factors:
In the case the equation is not exact, we can make it t b lti l i th ti ith i t ti exact by multiplying the equation with an integrating
factor σ(x) or σ(y) before solving (same steps as before)If (My‐Nx)/N is a solution in terms of x only( y x)/ y
If (My‐Nx)/M is a solution in terms of y only
Higher Order ODE’s2nd order homogeneous equation with constant coefficients
S k l i f h fSeek solution of the formObtain characteristic equation and solve for λ:
If λ1, λ2 are real & distinct solution is:
If λ1=λ2 (repeating roots), then the solution is:
If λ λ are complex i e of the form the solution is:If λ1, λ2 are complex i.e. of the form , the solution is:
Higher Order ODE’s (…cont)Solving for higher order equations is similar
Note that for k repeated roots, the linearly independent (LI) l ti i t d ith th t λ(LI) solutions associated with that λ are:
Higher Order ODE’s (…cont)Cauchy‐Euler Equation:
S k l ti f th fSeek solution of the formObtain characteristic equation and solve for λ:
If λ1, λ2 are real & distinct solution is:
If λ1=λ2 (repeating roots), then the solution is:
If λ λ are complex i e of the form the solution is:If λ1, λ2 are complex i.e. of the form , the solution is:
Higher Order ODE’s (…cont)Reduction of Order (for homogeneous non‐constant coefficients)If k l i f h ODE i If we know 1 solution for the ODE equation, we can use Lagrange’s method of variation of parameters to determine the second solutiondetermine the second solution…
LetLetThen we get:
Higher Order ODE’s (…cont)LetThen we get:This becomes a linear first order homogeneous ODESolve for p(x), integrate p(x) to solve for A(x)
Higher Order ODE’s (…cont)E.g. Legendre’s Equation
We can easily see that one solution would be Let the second solution be of the form: Plug in the solution into the ODE, simplify, and make th b tit ti A’( ) ( ) t tthe substitution A’(x) = p(x) to get:
Higher Order ODE’s (…cont)
Integrating p(x) and multiplying A(x) by y1(x) to get:
Higher Order ODE’s (…cont)And the general solution of the Legendre Equation is:
Higher Order ODE’s (…cont)2nd Order Non‐homogeneous Equations
Where L[y] is the nth order linear differential operatorf(x) is a linear combination of terms fk(x)G l l ti i t f th h l ti General solution consists of the homogeneous solution and particular solution:
Higher Order ODE’s (…cont)Method of Undetermined Coefficients
Two conditions:1. L is a linear, constant‐coefficient type2. Repeated differentiation of each fk(x) term results in only a finite number of LI termsUsed to find the particular solution y (x)Used to find the particular solution yp(x).
Steps:1. For each fk(x) term, determine the sequence consisting of itself and its derivatives2 The particular solution corresponding to fk(x) consists of a linear combination of the 2. The particular solution corresponding to fk(x) consists of a linear combination of the
LI terms found in the sequence generated3. If any terms are duplicated in the homogenous solution, multiply the entire linear
combination of terms by the lowest positive integer power of x so that no duplications are foundS b i h i l l i i h i L[ ] f ( )4. Substitute the particular solution into the equation L[y] = fk(x)
5. Solve for the undetermined coefficients6. The general solution is the sum of the homogeneous solution and all particular
solutions
Higher Order ODE’s (…cont)E.g.
Determine LI terms:
Thus, particular solutions will be of the form:
Higher Order ODE’s (…cont)Solve for the undetermined coefficients:
General solution:
Higher Order ODE’s (…cont)Method of Variation of Parameters
Not limited by the conditions imposed for method of d t i d ffi i tundetermined coefficients
Suppose you are given a second order ODE:
The homogeneous solution is:
Then we guess that the particular solution is similar to the homogeneous one by allowing the constants C1 and C t C2 to vary
Higher Order ODE’s (…cont)Derivation in your textbook…
Where