MindMap Gallery ordinary differential equations
Ordinary differential equations, including the solution of first-order differential equations, the solution of second-order reducible differential equations, and the solution of higher-order linear differential equations, let’s look at them together.
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This is a mind map about bacteria, and its main contents include: overview, morphology, types, structure, reproduction, distribution, application, and expansion. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about plant asexual reproduction, and its main contents include: concept, spore reproduction, vegetative reproduction, tissue culture, and buds. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about the reproductive development of animals, and its main contents include: insects, frogs, birds, sexual reproduction, and asexual reproduction. The summary is comprehensive and meticulous, suitable as review materials.
ordinary differential equations
The concept of differential equations
differential equations
Equations of the relationship between unknown functions and their derivatives and independent variables
ordinary differential equations
The unknown function is a differential equation of a function of one variable
order of differential equation
The order of the highest derivative in the equation
Solutions to Differential Equations
Substituting the function into the differential equation so that the equation becomes an identity, this function is the solution
General solutions to differential equations
If the solution of a differential equation contains the number of independent constants equal to the order of the differential equation, then the solution is a general solution of the differential equation.
Initial conditions and special solutions
Use concepts to solve problems
Solving First Order Differential Equations
variable separable
Such as:
convertible to variable separable type
Type like;
Homogeneous differential equations
mingle
First order linear differential equation
Such as:
General solution formula:
Notice:
Bernoulli equation
solution:
First transform it into:
Just solve this first-order linear differential equation
Solution of second-order reducible differential equations
Features: The unknown function y is not explicitly included in the equation
solution:
Features: The independent variable x is not explicitly included in the equation
solution:
Solving Higher Order Linear Differential Equations
concept
Second-order linear differential equation with variable coefficients
homogeneous equation
non-homogeneous equations
Second-order linear differential equation with constant coefficients
homogeneous equation
non-homogeneous equations
solution structure
####
but
like
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General solutions of second-order homogeneous linear differential equations with constant coefficients
Such as:
has two roots
General explanation:
have a root
General explanation:
There is a pair of conjugate roots:
General explanation:
Special solutions of second-order non-homogeneous linear differential equations with constant coefficients
Such as:
Special explanation:
k=
0, a is not a characteristic root
1. a is a single characteristic root
2. a is a double characteristic root
Special explanation:
k=
0
1
Solution of n-order homogeneous linear differential equation with constant coefficients