C₁e^(r₁t)+C₂e^(r₂t) gen solutions for the complex conjugate roots gen solutions for real & distinct roots La Place Transform Characteristic root (non-homogenous)

gen solutions for real & distinct roots

C₁e^(r₁t)+C₂te^(r₁t) gen solutions for the complex conjugate roots gen solutions for real repeated roots gen solutions for real & distinct roots La Place Transform

gen solutions for real repeated roots

e^(αt)(C₁cos(βt)+C₂sin(βt)) gen solutions for the complex conjugate roots gen solutions for real repeated roots Characteristic root (homogenous) Characteristic root (non-homogenous)

gen solutions for the complex conjugate roots

Diff Eq Workout Steps

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Calculus - Geometry

user_hassanuml Created by 6 mon ago

Cards in this deck(9)

1. Find the characteristic eqn 2. Define the roots of the characteristic eqn 3. Use the roots to build the basis solutions 4. Build the general solution 5. Use the initial conditions to build the specific solution

1. Use homogenous process to find the complementary Solution 2. Use variation of parameters or method of undetermined coefficient to determine the specific solution 3. Build the gen. soln. which is equal to the sum of the complementary solution & the particular solution 4. Use I.C.'s to build the specific solution

C₁e^(r₁t)+C₂e^(r₂t)

C₁e^(r₁t)+C₂te^(r₁t)

e^(αt)(C₁cos(βt)+C₂sin(βt))

1. Use the I.C.'s to transform the ODE into the S-domain 2. Solve the eqn in the s-domain for "u(s)" 3. Transform "u(s)" back into the time domain *the result is the specific solution w/respect to time*

1. Put the system in matrix form x' = Ax (w/"x" being a vector & A being a matrix) 2. Find eigenvalues & associated eigenvectors 3. Find the basis solution 4. Build the general solution 5. Use I.C.'s to build specific solution

1. Put system into matrix form x' = Ax+ f(t) (where "x" & "f(t)" are both vectors and A is a matrix) 2. Use homogenous step 2) to find basis solution 3. Build Phi = fundamental matrix 4. Build, particular soln = phi∫phi inverse * f(t)dt 5. Build, complimentary soln = phi [C₁; C₂;] 6. Build, gen. soln = complimentary + particular soln 7. Use I.C.'s to build specific solution

q₀ = 2 h = 0.1; t = 0:h:1 qstar = zeros(1,length(t)) qstar(1) = q₀ f=@(t,z)[f(t,z)] for ... end qstar(length(t)) qstar(11)

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