Characteristic Equations: Roots and Solutions of Linear Differential Equations

All flashcards

What is a characteristic equation?

An algebraic equation whose roots help construct solutions of suitable constant-coefficient differential or difference equations.

What does homogeneous mean for a differential equation?

The right-hand side is zero, as in $a_n y^{(n)}+\cdots+a_0y=0$.

Why is $y=e^{rx}$ useful as a trial function?

Its $j$th derivative is $r^j e^{rx}$, so substitution reduces the differential equation to a polynomial in $r$.

What is the characteristic equation for $a_n y^{(n)}+\cdots+a_0y=0$?

$a_n r^n+a_{n-1}r^{n-1}+\cdots+a_1r+a_0=0$.

What trial sequence is used for a constant-coefficient recurrence?

Try $y_t=r^t$ and substitute it into the recurrence to obtain its characteristic equation.

Quiz questions

1. A constant-coefficient differential equation has distinct real characteristic roots $r_1,\ldots,r_n$. Which form gives its general solution?Answer: $y(x)=\sum_{j=1}^{n}c_j e^{r_jx}$
2. A real characteristic root $r$ has multiplicity $k$. What form does its contribution to the general solution take?Answer: $e^{rx}$ multiplied by a polynomial of degree at most $k-1$
3. For real coefficients, what real-valued solution contribution comes from a complex-root pair $a\pm bi$, with $b\ne0$?Answer: $e^{ax}(C_1\cos bx+C_2\sin bx)$
4. For a constant-coefficient difference equation, what root condition gives stability in the criterion described?Answer: Every characteristic root has modulus less than $1$.
5. For a time-dependent, constant-coefficient linear differential equation, what root condition gives stability in the criterion described?Answer: Every characteristic root has a negative real part.