University Mathematics Handbook (2015)
XI. Ordinary Differential Equations, or ODE
Chapter 2. Linear n-th Order Differential Equations
2.1 Definition
a. The equation
(*)
is a -th order linear equation.
b. If the equation is called a homogeneous linear equation.
c. If all coefficients are continuous functions on , then in the neighborhood of initial conditions
, (**)
the Cauchy's problem (*, **) has a unique solution.
2.2 Linear Operator
a. We denote
b. For every fixed :
c.
d. If and are solutions of the equation , then is also a solution of that equation.
e. If ODE with real coefficients has complex solution , then and are real solutions of that equation.
f. If is a non-trivial solution of then, by substituting and , we get an order ODE for .
2.3 Solutions of
a. Functions are linearly dependent (LD) in interval if there exist numbers , not all of which are zeroes, such that . Otherwise, functions are linearly independent (LI).
b. If functions are LD on , then the determinant
is identically equal to zero.
Determinant is called Wronskian.
c. If LI functions are solutions of with continuous coefficients on , then Wronskian is zero nowhere on .
d. Particular solutions of , with a continuous coefficient, which are linear independent on , form a basis of the solution space.
e. Any solution of is a linear combination of LI solutions
2.4 Restoring Linear ODE
a. Let be LI functions in .
We construct an th order determinant
Developing it by its last column, we get a linear ODE the solutions of which are the given functions.
b. Abel's identity formula:
2.5 Linear Homogeneous ODE with Constant Coefficients
Let
be the n-th-order linear ODE, when are its constant real coefficients. Then, the solutions are in the form of . Substituting, we get the characteristic equation
a. If the characteristic equation has different real solutions , then are LI solutions forming a basis of solution space. The general solution is when are constants.
b. If the characteristic equation has complex solution , then is also one of its solutions. It is written . Therefore, this ODE has two LI solutions, and .
c. If a characteristic equation has a root with multiplicity , then it fits LI solutions .
d. If a characteristic equation has complex root with multiplicity , then it has real LI solutions:
2.6 Euler's Equation
The equation
when are real constant coefficients, is called Euler's equation.
a. The substitution transforms Euler's equation into a constant coefficient linear ODE.
b. We look for a solution in the form of . The result is a characteristic equation with respect to :
c. For each real root with multiplicity we get LI solutions
d. For each pair of complex roots with multiplicity we get LI solutions:
2.7 Non-homogeneous Linear ODE
a. The general solution of the ODE
(*)
is when is the general solution of the homogeneous equation and is a particular solution of the non-homogeneous equation, respectively.
b. The parameter variation method of finding a particular solution:
If is a general solution of (*) in the homogeneous equation, we look for a solution of the following form:
Substituting it in equation (*), we construct a linear equation system with respect to :
The determinant of the system is Wronskian . Therefore, the system has a unique solution.
2.8 Non-homogeneous Linear ODE with Constant Coefficients
(*)
Characteristic equation fitting (*)
(**)
If is one of the following functions:
1. Polynomial
2.
3. or
Then, we can find a particular solution of (*) on the form presented in the table, where is a given polynomial, and are unknown polynomials, and are real numbers:
Particular solution |
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Solutions |
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, solution with multiplicity |
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no solution |
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solution with multiplicity |
|
, no solution |
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solution with multiplicity |