Recipe

Solve a homogeneous second-order linear ODE

Use when: you have an equation of the form ay''+by'+cy=0 with constant coefficients a,b,c

Goal: the general solution y(x), built directly from the characteristic equation's roots

Taught in Distinct real roots · a later section extends this recipe with more cases

1. 1

   Replace y''\to r^2, y'\to r, and y\to1 to get ar^2+br+c=0.

   Example

   For y''-5y'+6y=0 (a=1,b=-5,c=6): r^2-5r+6=0.

   Why: substituting the trial solution y=e^{rx} into the ODE and dividing out e^{rx} — never zero — collapses the differential equation to this single algebraic one.

2. 2

   Solve ar^2+br+c=0 for r, factoring if you can and using the quadratic formula otherwise.

   Example

   r^2-5r+6=(r-2)(r-3)=0, so r=2 or r=3 — distinct and real.

   • Not yet: a repeated root — covered in Repeated real roots
   • Not yet: a pair of complex conjugate roots — covered in Complex roots and Euler's formula

   Why: the characteristic equation's roots are exactly the values of r for which e^{rx} solves the ODE — this section only covers the case where the two roots come out real and distinct.

3. 3

   With distinct real roots r_1\ne r_2, the general solution is y=C_1e^{r_1x}+C_2e^{r_2x}.

   Example

   y=C_1e^{2x}+C_2e^{3x}

   Why: distinct roots make e^{r_1x} and e^{r_2x} automatically linearly independent (this section's own Wronskian argument above), so Chapter 6's general-solution theorem applies with nothing further to check.

4. 4

   If you're given y(x_0) and y'(x_0), substitute x_0 into both y and y'=r_1C_1e^{r_1x}+r_2C_2e^{r_2x}.

   Example

   For y(0)=1, y'(0)=1: y(0)=C_1+C_2=1, and since y'=2C_1e^{2x}+3C_2e^{3x}, y'(0)=2C_1+3C_2=1.

   Why: this turns the two free constants into a system pinned down by the actual conditions of the problem, not just the equation.

5. 5

   Solve the resulting 2\times2 linear system for C_1,C_2, then substitute back into y.

   Example

   Substituting C_1=1-C_2 into 2C_1+3C_2=1 gives 2-2C_2+3C_2=1, so C_2=-1 and C_1=2. \boxed{y=2e^{2x}-e^{3x}}

   Why: the general solution is a whole family of curves; the conditions pick out the one curve that actually matches the problem.