Homogenous 2nd Order Linear with Constant Coefficients

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Table of Contents

• Homogenous with Constant Coefficients
  • Key Definitions
    • $f(\lambda )={\lambda }^2+p\lambda +q$
  • Homogenous 2nd Order Solutions
    • $y^{″}+p{y}^{\prime }+qy=0$
    • $f(\lambda )={\lambda }^2+p\lambda +q$
    • Distinct Real Roots
      • $y(t)=C_1{e}^{{\lambda }_{1}t}+C_2{e}^{{\lambda }_{2}t}$
    • Repeated Real Roots
      • $y(t)=C_1{e}^{{\lambda }_{1}t}+C_{2}t{e}^{{\lambda }_{1}t}$
    • Distinct Complex Roots
      • Complex Solution
        • $y(t)=C_1{e}^{{\lambda }_{1}t}+C_2{e}^{{\lambda }_{2}t}$
      • Real Solution
        • $y(t)=C_1{e}^{at}\cos bt+C_2{e}^{at}\sin bt$

#UCLA #Y1Q3 #Math33B

Homogenous with Constant Coefficients

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Key Definitions

Characteristic Polynomial - given $y^{\prime }‘+p{y}^{\prime }+qy=0$, 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

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Homogenous 2nd Order Solutions

Given diff. eq. of form:

$y^{″}+p{y}^{\prime }+qy=0$

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:

$y(t)=C_1{e}^{{\lambda }_{1}t}+C_2{e}^{{\lambda }_{2}t}$

Repeated Real Roots

If the char. pol. gives repeated, real roots, ${\lambda }_1\in \mathbb{R}$, then the general solution is:

$y(t)=C_1{e}^{{\lambda }_{1}t}+C_{2}t{e}^{{\lambda }_{1}t}$

Distinct Complex Roots

If the char. pol. gives repeated, real roots, ${\lambda }_1=a+bi,{\lambda }_2=a-bi$, then the general solutions are:

Complex Solution

$y(t)=C_1{e}^{{\lambda }_{1}t}+C_2{e}^{{\lambda }_{2}t}$

Real Solution

$y(t)=C_1{e}^{at}\cos bt+C_2{e}^{at}\sin bt$