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Higher Order Differential Equations - Coggle Diagram
Higher Order Differential Equations
No explicit dependence on variables
\( y = Ae^{\lambda x} \), RHS = 0
If repeated roots multiply second root by independent variable
If complex roots - represent as sines and cosines
Upon finding the complementary function, assume the particular function is a level higher than the complementary function but of the same form as the RHS - use the boundary conditions to evaluate their specific values
Explicit Variable Dependence
Euler and Legendre's Line equations
\( x = e^t \)
\(\dot y(x) = \frac{\alpha}{\alpha x + \beta} \dot y(t) \)
\(\ddot y(x) = \frac{\alpha^2}{(\alpha x + \beta)^2} (\ddot y(t) - \dot y(t) )\)
Exact Equations
Derivative of an ODE of one order lower
\( a_0(x) - a_1(x)' + a_2(x)''- ... + (-1)^n a_n(x) ^ n = 0 \)
Complementary Function Known
Variation of Parameters
Assume particular function of form \( y_p(x) = k_1(x)y_1(x) + k_2(x)y_2(x) + ...\)
Two main constraints
\( k_1'y_1(x) + k_2'y_2(x) = 0 \)
\( k_1'y_1(x)' + k_2'y_2(x)' = \frac{f(x)}{a_n(x)} \)
Green's Function
\( LG(x) = \delta (x-z) \) L = Linear differential operator
\( y(x) = \int_{a}^{ b} G(x,z)f(z) dz \)
Ly(x) = f(x) - comparing to subsitution
Properties of Green's function
Obeys original ODE w/ RHS = delta function
Obeys boundary conditions as a function of x
Derivatives upto order n-2 are continuous but n-1 has a finite discontinuity of \( 1/a_n(z) \)
Canonical Form for differential equations
A bit too convoluted to be really useful
General Order Differential Equations
Dependent Variable absent
\( p = \frac{dy}{dx}\)
Homogeneous equations
Assign weight m to y and n to x - if the combined weights of each term in an ODE is equal, we can assign a particular value for m then the eqn is isobaric \( y = vx^m, x = e^t \) leads to an eqn in which the new independent variable is absent
Power series solutions
Eigenvectors method of solving ode's