Homogeneous Linear Equation

Second Order

$$a(x)y^{\prime \prime }+b(x)y^{\prime }+c(x)y=0$$

• Note
  • $y=0$ is always a solution (the trivial solution)
  • A linear combination of any solutions is also a solution ($y(t)=c_1{y}_1(t)+c_2{y}_2(t)$)

Our focus will be on DEs with constant coefficients

$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$$

Solving

1. Find the characteristic/auxiliary/identity equation ($y^{\prime \prime }\to r^2,y^{\prime }\to r,y\to 1$)
2. Solve for roots
3. Turn the roots into solutions
   • Distinct Real Roots: $y_1=e^{r_{1}t},y_2=e^{r_{2}t}$
   • One Repeated Root: $y_1=e^{rt},y_2=t{e}^{rt}$
   • Complex Roots
     • $r=\alpha \pm i\beta$
     • $y_1=e^{\alpha t}\cos (\beta t),y_2=e^{\alpha t}\sin (\beta t)$
4. $y_h$ is linear combination of those

Higher Order

• Has Solutions in the form: $y_h(t)=c_1{y}_1(t)+c_2{y}_2(t)+\cdots +c_n{y}_n(t)$
• Solve the same way as second order just with more roots
• Often needs Rational Root Theorem