Crash introduction to infinitesimal calculus.
- Limits and Derivatives
- Integrals
- Ordinary Differential Equations - this file
A qualitative introduction to first and second order linear equations with constant coefficients is followed by the general process for solving linear equations with constant coefficients: the solution can be written as the sum of a free solution and a forced solution; the free solution immediately follows from the properties of the exponential function; the forced solution is computed exploiting some similarity with the the external forcing.
Examples of differential equations
The idea behind the general solution of linear ODEs with constant coefficients
Free solution
The free evolution of a system governed by a \(n^{th}\)-order linear ODE with constant coefficients, is the solution of the homogeneous (unforced, free) equation
\[y^{(n)}(t) + a_{n-1} y^{(n-1)}(t) + a_1 y'(t) + a_0 y(t) = 0 \ .\]
is a linear combination of (in general, complex) \(n\) exponential functions, \(e^{s_k t}\), for \(k=1:n\). In order to find all the values of the \(s_k\)s, the solution is assumed to be \(y(t) = C e^{s t}\). Its first derivative reads \(y'(t) = C s e^{st} = s y(t)\), its second derivative reads \(y''(t) = s^2 e^{st} = s^2 y(t)\), and so on. Replacing the expression of the derivatives in the homogeneous equation, and collecting and removing the never-zero factor \(y(t)\), an algebraic equation appears,
\[s^n + a_{n-1} s^{n-1} + \dots a_1 s + a_0 = 0 \ ,\]
whose \(n\) solutions (it’s a polynomial equation of degree \(n\), it has \(n\) complex solutions) are the values of the \(s_k\). The homogenoeus solution of the equation is thus a linear combination of the exponential functions,
\[y_{homo}(t) = \sum_{k=1}^{n} C_k e^{s_k t} \ .\]