• 4.1 Linear DEs; Fundamental Solutions; General Solutions
  • 4.2 Reduction of Order
  • 4.3 Homogeneous Linear DEs with Constant Coefficients
  • 4.4 Undetermined Coefficients – Superposition Approach
  • 4.5 Undetermined Coefficients – Annihilator Approach
  • 4.6 Variation of Parameters
  • 4.7 Cauchy-Euler Equations
  • 4.8 Solving Systems of Linear Equations by Elimination
  • 4.9 Nonlinear Equations

Homogeneous and Nonhomogeneous DEs

The first-order, linear DE \[ a_n\left(x\right) \frac{d^ny}{dx^n} + a_{n-1}\left(x\right) \frac{d^{n-1}y}{dx^{n-1}} + \cdots + a_1\left(x\right)\frac{dy}{dx} + a_0\left(x\right)y = g\left(x\right) \] is said to be homogeneous on an interval if $g$ is identically zero on that interval. It is said to be nonhomogeneous if it is not homogeneous.

Auxiliary Equation

The auxiliary equation of the nth-order homogeneous DE given in 4.1 is: \[ a_nm^n + a_{n-1}m^{n-1} + \cdots + a_2m^2 + a_1m + a_0 = 0 \] The roots of this equation comprise the general solution of the homogeneous DE in the following way:

  1. If $m$ is a real root of multiplicity $k$ then \[ c_1e^{mx}, \quad c_2xe^{mx}, \quad c_3x^2e^{mx}, \quad\cdots, \quad c_kx^{k-1}e^{mx} \] is a set of linearly independent solutions of the homogeneous DE and the linear combination \[ c_1e^{mx} + c_2 x e^{mx} + c_3 x^2 e^{mx} + \cdots + c_k x^{k-1} e^{mx} \] is therefore part of the general solution.
  2. If $m=\alpha + \beta i$ is a complex root of multiplicity $k,$ then \[ c_1 e^{\alpha x} \cos{\beta} x,\quad c_2 x e^{\alpha x} \cos{\beta} x,\quad c_3 x^2 e^{\alpha x} \cos{\beta} x,\quad \cdots,\quad c_k x^{k-1} e^{\alpha x} \cos{\beta} x \] is a set of linearly independent solutions of the homogeneous DE and the linear combination \[ c_1 e^{\alpha x} \cos{\beta} x + c_2 x e^{\alpha x} \cos{\beta} x + c_3 x^2 e^{\alpha x} \cos{\beta} x + \cdots + c_k x^{k-1} e^{\alpha x} \cos{\beta} x \] is part of the general solution.
  3. If $m=\alpha - \beta i$ is a complex root of multiplicity $k,$ then \[ c_1 e^{\alpha x} \sin{\beta} x,\quad c_2 x e^{\alpha x} \sin{\beta} x,\quad c_3 x^2 e^{\alpha x} \sin{\beta} x,\quad \cdots,\quad c_k x^{k-1} e^{\alpha x} \sin{\beta} x \] is a set of linearly independent solutions of the homogeneous DE and the linear combination \[ c_1 e^{\alpha x} \sin{\beta} x + c_2 x e^{\alpha x} \sin{\beta} x + c_3 x^2 e^{\alpha x} \sin{\beta} x + \cdots + c_k x^{k-1} e^{\alpha x} \sin{\beta} x \] is part of the general solution.

Since the auxiliary equation is an $n\mathrm{th}$ degree polynomial, the fundamental theorem of algebra guarantees that it has $n$ roots in the set of complex numbers, so these roots will be in one of above forms. Thus, the general solution of the homogeneous equation is the sum of all linear combinations that result from cases 1, 2, and 3 above.

Linear Non-Homogeneous Differential Equations Variation of Parameters

Procedure for Solving a Linear Non-homogeneous Differential Equation by Variation of Parameters

  1. Put the DE in the standard form \[ y^{\left(n\right)} + P_{n-1}\left(x\right)y^{\left(n-1\right)} + \cdots + P_1\left(x\right)y^\prime + P_0\left(x\right)y = f\left(x\right) \]
  2. Find the complementary solution $y_c.$
  3. Calculate the Wronskian of the fundamental set of solutions from 2.
  4. Let \[ u_1^\prime\left(x\right) = \frac{-y_2\left(x\right)f\left(x\right)}{W\left(s\right)} \\ \] and \[ u_2^\prime\left(x\right) = \frac{y_1\left(x\right)f\left(x\right)}{W\left(s\right)} \]
  5. Find $u_1$ and $u_2$ by integration.
  6. The particular solution is $y_p = u_1 y_1 + u_2 y_2.$
  7. The general solution is $y = y_c + y_p.$