6 Ordinary Differential Equations
A differential equation relates a function to its derivatives. Such equations arise whenever a law is stated as a rate — of growth, of cooling, of motion — which is to say almost everywhere in the physical sciences, so the ability to recover the function from the law is a central technique.
There is no single method. What exists is a classification, and the first skill is recognising which class an equation belongs to, since that determines the method. This section treats the first-order equations in turn — separable, homogeneous, linear, exact, and Bernoulli — with the test that identifies each and the procedure that solves it.
It then turns to linear equations of the second order, where the structure is different and more satisfying. Solutions superpose, and the general solution splits into a complementary function solving the reduced equation and a particular integral accounting for the forcing term: \(GS = CF + PI\). Two methods for the particular integral are given, undetermined coefficients and variation of parameters. The section closes with solution by power series, which reaches equations with variable coefficients that the earlier methods cannot touch.
Introduction
A differential equation is a a equation that involves one or more derivatives or differentials. Differential equations are classified by
- 1.
- types (namely, ordinary or partial)
- 2.
- order (i.e the highest \(-\) order derivative that occurs in the equation)
- 3.
- degree (the exponent of the highest order derivative)
\(\displaystyle {\Bigg (\dfrac {d^3y}{dx^3}\Bigg )^2 + \Bigg (\frac {d^2y}{dx^2}\Bigg )^5 + \frac {y}{x^2 + 1} = e^x}\) is an ordinary differentiable equation of order 3 and
degree 2.
Only \(``\)ordinary\(''\) derivatives occur when the dependent variable \(y\) is a function of one independent variable \(x\).
On the other hand, if the dependent variable is a function of two or more independent variables, say \(y=f(x,t)\), then partial derivatives of \(y\) many occur in such equation is called partial differential equations (PDEs). For example, \[\frac {\partial ^2y}{\partial t^2} = C^2\hspace {0.2cm} \frac {\partial ^2y}{\partial x^2}\hspace {0.2cm}\text {is a PDE}\]
Solution:
A function \(y = f(x)\) is said to be a solution of an ODE if the latter is satisfied when \(y\) and its derivatives are replaced throughout by \(f(x)\) and its derivatives. A differential equation of order \(n\) will generally have a solution involving \(n\) arbitrary constants. This is called the general solution.
6.1.1 Separable Equations
6.1.2 Homogeneous Equations
6.1.3 Linear Equations
6.1.4 Exact Equations
6.1.5 Integrating Factors
6.1.6 Bernoulli Equations
6.2 Linear Equations of the Second Order
6.2.1 Homogeneous Equations with Constant Coefficients
6.2.2 The Method of Undetermined Coefficients
6.2.3 The Method of Variation of Parameters
6.3 Solution by Series
6.4 Practice Problems
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