Differential Equation

Linear: dy/dt+p(t)y=f(t)

Solve Using the Integrating Factor Method

y=(e^-∫p(t)dt)+∫(f(t)e^∫p(t)dt)dt+ce^-∫p(t)dt

Use IVP to solve for c

Solve Using the Euler-Lagrange Method

Find the solutions to the homogeneous equation: yh=ce^-∫p(t)dt

Solve: v'(t)e^-∫p(t)dt=f(t) for v(t)

Obtain a particular solution: yp=v(t)e^-∫p(t)dt

Combine the solutions to the homogeneous equation and the particular solution: y(t)=yh+yp

Use IVP to solve for c

Perform a Laplace transform of f(t) to get F(s)

Perform a Laplace transform of the equation to get Y(s)

Use the inverse Laplace transform on F(s) and Y(s) to get y(t)

Solve y(t) by using the IVP

Nonlinear

Qualitative Analysis

Logistic Equation

y'=r(L-y/L)y, where r is called the initial growth rate, and L is called the carrying capacity

Linear

f(t) equal to zero

Homogeneous

Constant Coefficients: ay"+by'+cy=0

Δ=b^2-4ac

For Δ>0: r1=(-b+√∆)/2a and r2=(-b-√∆)/2a

y(t)=c1e^(r1t)+c2e^(r2t)

Use IVP to solve for c1 and c2

For Δ=0: r=-b/2a

y(t)=c1e^(rt)+c2te^(rt)

For Δ<0: r1,r2=α±βi, α=-b/2a, β=√(-∆)/2a

y(t)=(e^αt)(c1cosβt+c2sinβt)

Use IVP to solve for c1 and c2

Undamped Harmonic Oscillator:mẍ+bẋ+kx=0, where b=0

x(t)=c1 cosω0 t+c2 sinω0 t, where ω0=√(k/m)

Use IVP to solve for c1 and c2

Two-dimensional system: x'=Ax, Where A is a matrix of constants.

Find Eigenvalues(λ1 and λ2) and Eigenvectors(v1 and v2)

Real Eigenvalues

x(t)=c1e^(λ1t)v1+c2e^(λ2t)v2

Use IVP to solve for c1 and c2

Nonreal Eigenvalues

λ1 and λ2 are of the form α±βi v1 and v2 are of the form p±iq

Xre=e^αt(cosβtp-sinβtq) Xim=e^αt(sinβtp+cosβtq)

x(t)=c1Xre(t)+c2Xim(t)

Use IVP to solve for c1 and c2

f(t) not equal to zero

Nonhomogeneous

Constant Coefficients: ay"+by'+cy=f(t)

Δ=b^2-4ac

For Δ>0: r1=(-b+√∆)/2a and r2=(-b-√∆)/2a

yh=c1e^(r1t)+c2e^(r2t)

If f(t) is in Exp Family: for example, y"-y'-2y=2e^-3t

Then yp=Ae^-3t

Solve for A: Find yp', plug it in for y', then find yp", plug it in for y", plug yp in for y.

y(t)=yh+yp

Use IVP to solve for c1 and c2

If f(t) is in Trig Family: for example, y"-y'-2y=2cos3t

Then yp=Acos3t+Bsin3t

Solve for A and B: Find yp', plug it in for y', then find yp", plug it in for y", plug yp in for y. Use two equations and two unknowns.

y(t)=yh+yp

Use IVP to solve for c1 and c2

If f(t) is in Polynomial Family: for example, y"-y'-2y=3t^2-1

Then yp=At^2+Bt+C

Solve for A, B, and C: Find yp', plug it in for y', then find yp", plug it in for y", plug yp in for y. Equate coefficients.

y(t)=yh+yp

Use IVP to solve for c1 and c2

If f(t) has the form of two families, we mix the two: for example, y"-y'-2y=(t^2)e^t

Then yp=(At^2+Bt+C)e^t

Solve for A, B, and C: Find yp', plug it in for y', then find yp", plug it in for y", plug yp in for y. Equate coefficients.

y(t)=yh+yp

Use IVP to solve for c1 and c2

Find yp by variation of parameters

yp=v1y1+v2y2, where v1'=-y2f/W(y1,y2) and v2'=y1f/W(y1,y2)

y(t)=yh+yp

Use IVP to solve for c1 and c2

For Δ=0: r=-b/2a

yh=c1e^(rt)+c2te^(rt)

Find yp the same way as for Δ>0

y(t)=yh+yp

Use IVP to solve for c1 and c2

Find yp by variation of parameters

yp=v1y1+v2y2, where v1'=-y2f/W(y1,y2) and v2'=y1f/W(y1,y2)

y(t)=yh+yp

Use IVP to solve for c1 and c2

For Δ<0: r1,r2=α±βi, α=-b/2a, β=√(-∆)/2a

yh=(e^αt)(c1cosβt+c2sinβt)

Find yp the same way as for Δ>0

y(t)=yh+yp

Use IVP to solve for c1 and c2

Find yp by variation of parameters

yp=v1y1+v2y2, where v1'=-y2f/W(y1,y2) and v2'=y1f/W(y1,y2)

y(t)=yh+yp

Use IVP to solve for c1 and c2

Perform a Laplace transform of f(t) to get F(s)

Perform a Laplace transform of the equation to get Y(s)

Use the inverse Laplace transform on F(s) and Y(s) to get y(t)

Solve y(t) by using the IVP

Nonlinear

Linearize

Analyze a related linear system of an autonomous nonlinear system near an equilibrium

Find numerical solution through Euler's Method

Linear

Perform a Laplace transform of f(t) to get F(s)

Perform a Laplace transform of the equation to get Y(s)

Use the inverse Laplace transform on F(s) and Y(s) to get y(t)

Solve y(t) by using the IVP

Nonlinear

Linearize

Analyze a related linear system of an autonomous nonlinear system near an equilibrium

Find numerical solution through Euler's Method