20 d final formulas

8/12/2019 20 d Final Formulas

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Formula Sheet for Final Exam

• Separable:

M (x) + N (y)dydx

= 0 , i.e. dy

dx = f (y)g(x)

Separate to get dyf (y) = g(x)dx, then integrate both sides.

• First-Order Linear:dydt

+ p(t)y = g(t)

Multiply both sides by the integrating factor µ(t) = eR p (t )dt , then integrate.

• Autonomous:dydt

= f (y)

Equilibrium solutions correspond to roots of f (y). We can sketch solutions by hand.

• Exact:M (x, y) + N (x, y)

dydx

= 0 , where M y = N x

General solution is given implicitly by ψ(x, y ) = C , where ψ satises

∂ψ∂x

= M and ∂ψ

∂y = N.

• Second-Order Linear Homogeneous Constant Coefficients:

ay + by + cy = 0

If r1 = r2 are real roots of the characteristic polynomial ar 2 + br + c, then the generalsolution is

y(t) = c1er 1 t + c2er 2 t .

• Wronskian:W (y1, y2)( t) = y1(t)y2(t) − y2(t)y1(t)

y1 and y2 are linearly independent on an open interval I if and only if there is some t0 ∈ I such that W (y1, y2)( t0) = 0.

• Second-Order Linear Homogeneous Constant Coefficient Differential Equa-tions:

ay + by + cy = 0 , a = 0– If r 1 and r2 are distinct real roots of the characteristic equation ar 2 + br + c = 0, then

the general solution is y(t) = c1er 1 t + c2er 2 t .– If r = α ± iβ are complex roots of the characteristic equation, then the general solution

is y(t) = c1eαt cos(βt ) + c2eαt sin(βt ).– If the characteristic equation has a repeated root r, then the general solution is

y(t) = c1ert + c2te rt .



• Reduction of Order: If y1 is a solution of the second-order linear homogeneous equation

y + p(t)y + q (t)y = 0

then we guess y2(t) = v(t)y1(t), where v(t) is some unknown function of t. Plugging thisexpression for y2 back into the equation leads to a rst-order linear equation for v , whichwe can solve using integrating factors (or possibly separation of variables). Finally, wetake the antiderivative to nd v, and multiply by y1 to get y2.

• Undetermined Coefficients: To nd a particular solution Y p(t) of the non-homogeneousequation ay + by + cy = g(t), we guess a function that “looks like” the most generalfunction that we could form from g(t) and its derivatives. A table of guesses for the mostcommon functions g(t) is given below.

g(t) Y p(t)P (t) ts (An tn + An − 1tn − 1 + . . . + A1t + A0)

P (t)ert t s (An tn + An − 1tn − 1 + . . . + A1t + A0)ert

P (t)cos(βt )P (t)sin( βt ) ts

(An tn + An −

1tn − 1 + . . . + A

1t + A

0)cos(βt )

+( Bn tn + Bn − 1tn − 1 + . . . + B1t + B0)sin( βt )P (t)eαt cos(βt )P (t)eαt sin(βt ) ts (An tn + An − 1tn − 1 + . . . + A1t + A0)eαt cos(βt )

+( Bn tn + Bn − 1tn − 1 + . . . + B1t + B0)eαt sin(βt )

(Here P (t) is a polynomial of degree n, and s is the smallest power of t so that Y p(t) isnot a solution of the corresponding homogeneous equation.)

• Variation of Parameters: If y1 and y2 are solutions of the homogeneous equationy + p(t)y + q (t)y = 0, then a particular solution Y p(t) of the non-homogeneous equationy + p(t)y + q (t)y = g(t) is given by

Y p(t) = u1(t)y1(t) + u2(t)y2(t),

where u1 and u2 satisfy

u1 = − y2g(t)W (y1, y2)

u2 = y1g(t)W (y1, y2)

.

(Here W (y1, y2) = y1y2 − y2y1 is the Wronskian of y1 and y2.)

• First-Order Two-Dimensional Linear Homogeneous Constant Coefficient Sys-tems:

x = Ax

– If λ1 and λ2 are distinct real eigenvalues of A, with corresponding eigenvectors v1and v2, then the general solution is x(t) = c1 v1eλ 1 t + c2 v2eλ 2 t .

– If a ± ib are complex conjugate eigenvalues of A, then the general solution can befound by nding the complex eigenvector v corresponding to a + ib and using the realand imaginary parts of x(t) = ve(a + ib ) t as a fundamental set of solutions.



– If λ is a repeated eigenvalue for A, then the general solution is x(t) = c1 veλt +c2 (vt + ρ) eλt , where v is the eigenvalue for λ , and ρ satises (A − λI ) ρ = v.

• Series Solutions: To nd a series solution of the differential equation p(x)y + q (x)y +r (x)y = 0 about an ordinary point x0, plug the series

y(x) =∞

n =0an (x − x0)n

into the differential equation to nd the recurrence relation satised by the an . The radiusof convergence of the series solution is bounded below by the minimum of the radii of convergence of the series for q/p and r/p . Note that any rational function in reducedform f /g has a convergent power series about x0 as long as g(x0) = 0, and its radius of convergence is the distance between x0 and the nearest zero of g in the complex plane.

• Laplace Transforms: To solve a linear constant coefficient differential equation of theform

an y(n )

+ an−

1y(n − 1)

+ · · · a2y + a1y + a0y = g(t)with given initial conditions, take the Laplace Transform of both sides, solve the resultingequation for Y (s), and then take the inverse Laplace transform to get the solution y(t).

L{1} = 1s L{eat } = 1

s − a L{tn } = n !s n +1

L{sin(at )} = as 2 + a 2 L{cos(at )} = s

s 2 + a 2 L{sinh( at )} = as 2 − a 2

L{cosh(at )} = ss 2 − a 2 L{eat sin(bt)} = b

(s − a )2 + b2 L{eat cos(bt)} = s − a(s − a )2

L{tn eat } = n !(s − a ) n +1 L{uc(t)} = e− cs

s L{uc(t)f (t − c)} = e− cs

L{δ (t − c)} = e− cs L{f (n ) (t)} = sn F (s) − sn − 1f (0) − · · · − f (n − 1) (0)

20 d final formulas

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