First-order differential equation

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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:

F(x,y,y') = 0

where F is an expression (function) involving three variables. Note that F must make use of y' (also written as dy/dx), but it could ignore x or y.

The theory and terminology follows that for the general concept of differential equation.

Solution concept

  • Functional solution: A function f on the domain of interest is said to be a solution (or functional solution) to the equation if, when we plug in y = f(x), the equation holds true for all x in the domain, i.e.:

F(x,f(x),f'(x)) = 0 \ \forall \ x \in \operatorname{dom}(f)

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 R(x,y) = 0 is termed a relational solution to the equation if F(x,y,y') = 0 holds true for all x,y if we calculate the derivative y' using implicit differentiation.

Initial value problem

An initial value problem is a first-order differential equation

F(x,y,y') = 0

along with a point (x_0,y_0).

A functional solution to the initial value problem is a solution y = f(x) to the first-order differential equation such that f(x_0) = y_0.

A relational solution to the initial value problem is a solution R(x,y) = 0 to the first-order differential equation such that R(x_0,y_0) = 0.

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 y = f(x) is termed a functional solution and a solution in the form of a relation R(x,y) = 0 is termed a relational solution. y = \sin x is a functional solution to y^2 + y'^2 = 1.
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. y = \sin(x + C) with parameter C \in \R, is a solution family for y^2 + y'^2 = 1.
general solution a solution family that covers all solutions (or almost all solutions, possibly excluding some exceptions) The general solution to y' = 0 is y = C, C \in \R.
solution to initial value problem a particular solution that satisfies the initial value condition. A particular solution to y + y' = 0 satisfying y(0) = 1 is y = e^{-x}.

Facts

  • As a general principle, the way to solve a first-order differential equation is to convert it to an integration problem. The additive +C 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 xy-plane corresponding to any solution to the differential equation. For instance, if R(x,y) = 0 is a relational solution, then the curve R(x,y) = 0 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 n, we generally expect up to n 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 f 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 y' + p(x)y = q(x) 1 Use the integrating factor e^{H(x)} where H'=p. The general solution is y = Ce^{-H(x)} + e^{-H(x)}\int p(x)e^{H(x)} \, dx
separable differential equation which is of the form y' = f(x)g(y) (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 \int \frac{dy}{g(y)} = \int f(x) \, dx. Also find solutions corresponding to y = k where g(k) = 0.
first-order exact differential equation F(x,y,y') = 0 1 Try to find a relation R(x,y) such that F(x,y,y') = \frac{d}{dx}[R(x,y)] using implicit differentiation. Finding the R, even if it does exist, can be tricky.
Bernoulli differential equation y' + p(x)y = q(x)y^n (n \ne 0,1) 1 Divide both sides by y^n (set aside possible stationary solution y = 0), then substitute w = 1/y^{n-1} to get a first-order linear differential equation with dependent variable w and independent variable x.
Clairaut's equation which is of the form y = xy' + f(y') need not be polynomial; if polynomial, may have any degree y = Cx + f(C) with C \in \R (all straight lines) and a single other solution explicitly described as the solution to x + f(dy/dx) = 0, given by x = -f'(p), y = f(p) - pf'(p) as a parametric curve in terms of p.
Lagrange equation y = f(y')x + g(y') which is linear in x and y but not necessarily in y' 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 y' (which we denote by p). There may be some special straight line solutions of the form y = px + g(p) for values p with p = f(p).

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 u = g(x,y) and replace one of the variables x,y with u. Then, after solving, we plug back u = g(x,y) to get the relational solutions between x and y.
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.