Homogenous 2nd Order Linear with Constant Coefficients
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Table of Contents
- Homogenous with Constant Coefficients
#UCLA #Y1Q3 #Math33B
Homogenous with Constant Coefficients
Key Definitions
Characteristic Polynomial - given $y’‘+py’+qy=0$, the char. pol. is:
\(f(\lambda)=\lambda^2+p\lambda+q\)
s.t. the roots are called the characteristic roots Note: the discriminant of the quadratic eq. of the char. pol. can be distinct-real, same-real, or distinct-complex
Homogenous 2nd Order Solutions
Given diff. eq. of form:
\(y''+py'+qy=0\)
and char. pol.:
\(f(\lambda)=\lambda^2+p\lambda+q\)
having roots of 3 different outcomes:
Distinct Real Roots
If the char. pol. gives distinct, real roots, $\lambda_1,\lambda_1\in\mathbb R$, then the general solution is:
\(y(t)=C_1e^{\lambda_1t}+C_2e^{\lambda_2t}\)
Repeated Real Roots
If the char. pol. gives repeated, real roots, $\lambda_1\in\mathbb R$, then the general solution is:
\(y(t)=C_1e^{\lambda_1t}+C_2te^{\lambda_1t}\)
Distinct Complex Roots
If the char. pol. gives repeated, real roots, $\lambda_1=a+bi,\lambda_2=a-bi$, then the general solutions are: