Math 221 - Review sheet for Exam 1
First order equations
Separable equations can be written in the form f (y) · dy/dt = g(t), with solution
Z
Z
f (y) dy = g(t)dt + C.
Linear equations can be written in the form y 0 + p(t)y = g(t). To solve, put
R
µ(t) = e
p(t)dt
.
Multiply through by µ to get
d
(µy) = µg,
dt
Integrate and divide by µ to get
−1
y(t) = µ (t)
Z
t
µ(s)g(s)ds + Cµ−1 (t).
Modeling problems examined include exponential growth and decay, logistic growth, Newton’s law of cooling, simple problems with Newton’s 2nd law, including computing the
escape velocity.
Autonomous ODEs have the form dy/dt = f (y); the right hand side is independent of t.
Complete qualitative information about the solutions can be obtained by finding the
equilibrium points (those values of y at which f (y) = 0) and determining their stability
(use the sign of f to determine if nearby points are attracted or repelled
Direction fields: For the ODE y 0 = f (t, y), draw a small line segment at the point (t, y)
with slope f (t, y). The “field’ of line segments is called the direction field for the ODE.
The solutions to the ODE consist of the curves that are everywhere tangent to the
direction field.
Euler’s method gives a numerical approximation to the IVP y 0 = f (t, y); y(a) = y0 on the
time interval [a, b]. Divide the interval into n equal parts of length ∆t = (b − a)/n.
Then the algorithm is given by the recursion relation
yk = yk−1 + f (tk−1 , yk−1 )∆t, for k = 1, 2, . . . , n,
where tk = a + k∆t, and t0 = a. The approximate solution is obtained by connecting
the points (t0 , y0 ), (t1 , y1 ), . . . , (tn , yn ) with straight line segments.
Existence/uniqueness theorem: If the function f (t, y) is differentiable in some rectangle
in the t−y plane, and (t0 , y0 ) lies in the rectangle, then the IVP y 0 = f (t, y); y(t0 ) = y0
has a unique solution in the rectangle.
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Second order linear equations
General material: The general 2nd order linear differential equation is of the form
y 00 + p(t)y 0 + q(t)y = g(t)
Any point at which p or q fails to be continuous is called a singular point of the ODE.
In any time interval in which p, q, and g are continuous, the initial value problem
y 00 + py 0 + qy = g; y(t0 ) = y0 , y 0 (t0 ) = y00 has a unique solution for all t in the interval.
The ODE can be written in the form L[y] = g, where, for any function y, the differential
operator L takes y and produces the new function L[y] = y 00 + py 0 + qy. L is called a
linear operator because of the following fundamental property: if y1 and y2 are any 2
differentiable functions, and c1 , c2 are arbitrary real numbers, then
L[c1 y1 + c2 y2 ] = c1 L[y1 ] + c2 L[y2 ].
This is particularly important for the solutions of the homogeneous equation L[y] = 0.
It says that if y1 and y2 are two solutions to L[y] = 0, then for any constants c1 , c2 ,
the function c1 y1 (t) + c2 y2 (t) is also a solution. This is also called the superposition
principle. Many important equations in physics and engineering are linear in this sense,
including Laplace’s equation, Maxwell’s equations (source-free), the heat equation,
the wave equation, and Schrodinger’s equation. These all satisfy the superposition
principle.
Fundamental sets: A pair of functions {y1 (t), y2 (t)} is said to form a fundamental set of
solutions to the homogeneous equation L[y] = 0 if any solution to the IVP can be
written in the form y = c1 y1 + c2 y2 for some choice of constants.
A necessary and sufficient condition for two solutions {y1 (t), y2 (t)} to form a fundamental set on the interval a < t < b is that the Wronskian
y1 (t) y2 (t)
W (y1 , y2 )(t) = det
6= 0
y10 (t) y20 (t)
R
at any point t0 in the interval. Since W (t) = C exp(− p(t)dt), it is either identically
0 or it’s never 0 in the interval.
If {y1 (t), y2 (t)} form a fundamental set of solutions to L[y] = 0, then the general
solution to the homogeneous equation is y = c1 y1 + c2 y2 , with c1 , c2 arbitrary. This
means that the IVP can be solved by finding the unique values of the constants such
that
y0 = c1 y1 (t0 ) + c2 y2 (t0 )
y00 = c1 y10 (t0 ) + c2 y20 (t0 )
Fundamental sets are not unique: if {y1 , y2 } is one fundamental set, so is {ŷ1 = ay1 +
by2 , ŷ2 = cy1 + dy2 } for any real numbers a, b, c, d such that ad − bc 6= 0.
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Constant coefficients: The 2nd order linear ODE is said to have constant coefficients if
it takes the form ay 00 + by 0 + cy = g(t), where a, b, and c are constants. In this case,
the homogeneous equation (g = 0) has solutions of the form y(t) = ert . The possible
values of r can be found by solving the characteristic equation
ar2 + br + c = 0.
There are three possible cases:
1. The two roots are real and distinct, r1 6= r2 . The general solution is
y = c1 er1 t + c2 er2 t .
2. The two roots are complex conjugates of one another: r± = a ± ib. Then a
fundamental set is formed by the real and imaginary parts of z = e(a+ib)t . The
general solution is
y = eat (c1 cos(bt) + c2 sin(bt)).
3. There is a repeated root: r1 = r2 = r. In this case, y1 (t) = ert is one solution,
and another is given by y2 (t) = tert . So the general solution is
y = (c1 + c2 t)ert .
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