First-order differential equation
Contents
Definition
Formal description
The term first-order differential equation is used for any differential equation whose order is 1. In other words, it is a differential equation of the form:
where is an expression (function) involving three variables. Note that must make use of (also written as ), but it could ignore or .
The theory and terminology follows that for the general concept of differential equation.
Solution concept
- Functional solution: A function on the domain of interest is said to be a solution (or functional solution) to the equation if, when we plug in , the equation holds true for all in the domain, i.e.:
Note that in cases of functions defined on closed intervals, we exclude checking the conditions on the boundary of the domain because two-sided derivatives don't make sense at the boundary.
- Relational solution: A relation is termed a relational solution to the equation if holds true for all if we calculate the derivative using implicit differentiation.
Initial value problem
An initial value problem is a first-order differential equation
along with a point .
A functional solution to the initial value problem is a solution to the first-order differential equation such that .
A relational solution to the initial value problem is a solution to the first-order differential equation such that .
Terminology
Solution terminology
Term | Meaning | Example |
---|---|---|
particular solution | a function or relation that is a solution for the equation (see #Solution concept). A solution in the form of a function is termed a functional solution and a solution in the form of a relation is termed a relational solution. | is a functional solution to . |
solution family | a family of functions or relations, with one or more parameters possibly subject to some constraints, such that for every choice of parameter values subject to those constraints, we get a particular solution. | with parameter , is a solution family for . |
general solution | a solution family that covers all solutions (or almost all solutions, possibly excluding some exceptions) | The general solution to is . |
solution to initial value problem | a particular solution that satisfies the initial value condition. | A particular solution to satisfying is . |
Facts
- As a general principle, the way to solve a first-order differential equation is to convert it to an integration problem. The additive appearing in the indefinite integration gives the freely varying parameter for the solution family.
- As a general principle, the number of degrees of freedom (i.e., the number of independent freely varying parameters) in the general solution to a first-order differential equation is 1. In other words, we expect the solution space to be one-dimensional.
- As a general principle, the number of solutions to an initial value problem should be finite. If the differential equation is nice enough, then there should be a unique solution to any initial value problem.
Geometric description of solutions
This geometric description is qualitatively different for first-order differential equations compared to higher-order differential equations.
A solution curve to a differential equation is a curve in the -plane corresponding to any solution to the differential equation. For instance, if is a relational solution, then the curve gives a solution curve for the differential equation.
For first-order differential equations, we generally expect that specifying a point on the curve uniquely determines the solution curve (because that's an initial value specification). Thus, the solution curves are expected to be (mostly) non-intersecting curves that (hopefully) cover the plane or a large part of the plane. This general expectation is likely to be met for first-order first-degree differential equations. For first-order differential equations of degree , we generally expect up to solution curves through a generic point.
Solution strategies
Solution strategies in particular cases
Below are some formats of equations for which general strategies are known. Note that the letter is no longer used for the solution function but may be used for other functions.:
Equation type | Degree (if polynomial in highest order derivative) | Quick summary of solution strategy |
---|---|---|
first-order linear differential equation which in simplified form looks like | 1 | Use the integrating factor where . The general solution is |
separable differential equation which is of the form (any first-order first-degree autonomous differential equation is separable, though there are separable differential equations that aren't autonomous) | 1 | Separate and solve as . Also find solutions corresponding to where . |
first-order exact differential equation | 1 | Try to find a relation such that using implicit differentiation. Finding the , even if it does exist, can be tricky. |
Bernoulli differential equation () | 1 | Divide both sides by (set aside possible stationary solution ), then substitute to get a first-order linear differential equation with dependent variable and independent variable . |
Clairaut's equation which is of the form | need not be polynomial; if polynomial, may have any degree | with (all straight lines) and a single other solution explicitly described as the solution to , given by as a parametric curve in terms of . |
Lagrange equation which is linear in and but not necessarily in | need not be polynomial; if polynomial, may have any degree | General solution is a family of curves, each described as a parametric curve with parameter the derivative (which we denote by ). There may be some special straight line solutions of the form for values with . |
Reduction methods
The following are methods that may be used to reduce more complicated first-order differential equations to simpler ones:
Method | Can it reduce the degree of the differential equation, and potentially convert a higher degree differential equation to first-degree differential equations? | Description |
---|---|---|
substitution method for solving differential equations (the substitutions of interest are zeroth-order substitutions) | No | We put and replace one of the variables with . Then, after solving, we plug back to get the relational solutions between and . |
factorization method for solving differential equations | Yes | Bring everything to one side, factor it as a product of two expressions, then solve each as a separate differential equation, and combine solutions. Also, check for mixed solutions. |