Recall: A differential equation (often called a “de”) is an equation involving derivatives of an unknown function. If the unknown can be assumed to be a function of only one variable (so the derivatives are the “ordinary” derivatives from Calc. I), then we say the differential equation is an ordinary differential equation (ode). Otherwise, the equation is a partial differential equation (pde). Our interest will just be in odes. In these notes, the variable will usually be denoted by x and the unknown function by y or y(x ) .
Recall, also, for any given ordinary differential equation:
1. The order is the order of the highest order derivative of the unknown function explicitly appearing in the equation.
2. A solution is any function (or formula for a function) that satisfies the equation.
3. A general solution is a formula that describes all solutions to the equation. Typically, the general solution to a kth order ode contains k arbitrary/undetermined constants.
4. Typically, a “differential equation problem” consists of a differential equation along with some auxiliary conditions the solution must also satisfy (e.g., “initial values” for the solution). In practice you usually find the general solution first, and then choose values for the “undetermined constants” so that the auxiliary conditions are satisfied.
1.2 Some “Analytic” Methods for Solving First-Order ODEs
(Warning: Here, the word “analytic” just means that the method leads to exact formulas for solutions, as opposed to, say, a numerical algorithm that gives good approximations to particular solutions at fixed points. Later in this course, the word “analytic” will mean something else.)
Separable Equations
A first-order ode is separable if it can be written as
Such a de can be solved by the following procedure:
1. Get it into the above form (i.e., the derivative equaling the product of a function of x (the g(x ) above), with a function of y (the above h(y) ).
2. Divide through by h(y) (but also consider the possibility that h(y) = 0 ).
3. Integrate both sides with respect to x (don’t forget an arbitrary constant).
4. Solve the last equation for y(x ) .
Linear Equations
A first-order ode is said to be linear if it can be written in the form
where p(x ) and q (x ) are known functions of x . Such a differential equation can be solved by the following procedure:
1. Get it into the above form.
2. Compute the integrating factor
(don’t worry about arbitrary constants here).
3. (a) Multiply the equation from the first step by the integrating factor.
(b) Observe that, by the product rule, the left side of the resulting equation can be rewritten as dd x [μy] , thus giving you the equation
4. Integrate both sides of your last equation with respect to x , and solve for y(x ) . Don’t forget the arbitrary constant.
Two notes on this method:
1. The formula for the integrating factor μ(x ) is actually derived from the requirement that
which is the “observation” made in step 3b of the procedure. This means that μ must satisfy the simple differential equation
2. Many texts state a formula for y(x ) in terms of p(x ) and q (x ) . The better texts also state that memorizing and using this formula is stupid.
Other Methods
Other methods for solving first-order ordinary differential equations include the integration of exact equations, and the use of either clever substitutions or more general integrating factors to reduce “difficult” equations to either separable, linear or exact equations. See a good de text if you are interested.
1.3 Higher-Order Linear Differential Equations Basics
An N th order differential equation is said to be linear if it can be written in the form
where f and the ak ’s are known functions of x (with a0 (x ) not being the zero function). The equation is said to be homogeneous if and only if f is the zero function (i.e., is always 0 ).
Recall that, if the equation is homogeneous, then we have “linearity”, that is, whenever y1 and y2 are two solutions to a homogeneous linear differential equation, and a and b are any two constants, then y = ay1 + by2 is another solution to the differential equation. In other words, the set of solutions to a homogeneous linear differential equations is a vector space of functions. (Isn’t it nice to see vector spaces again?)
Recall further, that
1. The general solution to an N th order linear homogeneous ordinary differential equation is given by
where the ck ’s are arbitrary constants and
{y1, y2, . . . , yN }
is a linearly independent set of solutions to the homogeneous de. (i.e., {y1, y2, . . . , yN } is a basis for the N -dimensional space of solutions to the homogeneous differential equation.)
2. A general solution to an N th order linear nonhomogeneous ordinary differential equation is given by
where y p is any particular solution to the nonhomogeneous ordinary differential equation and yh is a general solution to the corresponding homogeneous ode.
In “real” applications, N is usually 1 or 2 . On rare occasions, it may be 4 , and, even more rarely, it is 3 . Higher order differential equations can arise, but usually only in courses on differential equations. Do note that if N = 1 , then the differential equation can be solved using the method describe for first order linear equations
Second-Order Linear Homogeneous Equations with Constant Coefficients
Consider a differential equation of the form
where a , b , and c are (real) constants. To solve such an equation, assume a solution of the form
(where r is a constant to be determined), and then plug this formula for y into the differential equation. You will then get the corresponding characteristic equation for the de,
Solve the characteristic equation. You’ll get two values for r ,
(with the possibility that r+ = r? ). Then:
1. If r+ and r? are two distinct real values, then the general solution to the differential equation is
where c1 and c2 are arbitrary constants.
2. If r+ = r? , then r+ is real and the general solution to the differential equation is
where c1 and c2 are arbitrary constants. (Note: The c2 x er+ x part of the solution can be derived via the method of “reduction of order”.)
3. If r+ or r? is complex valued, then they are complex conjugates of each other,
for some real constants α and β . The general solution to the differential equation can then be written as
where c1 and c2 are arbitrary constants. However, because
the general solution to the differential equation can also be written as
where C1 and C2 are arbitrary constants. In practice, the later formula for y is usually
preferred because it involves just real-valued functions.
Second-Order Euler Equations
A second-order Euler equation2 is a differential equation that can be written as
where a , b , and c are (real) constants. To solve such an equation, assume a solution of the form
(where r is a constant to be determined), and then plug this formula for y into the differential equation, and solve for r .
With luck, you will get two distinct real values for r , r1 and r2 , in which case, the general solution to the differential equation is
where c1 and c2 are arbitrary constants.
With less luck, you only complex values for r , or only one value for r . See chapter 16 of the online text to see what to do in these cases.
