Differential equations and dynamical systems

Linear differential equations

Higher-order equations with a superposition principle, solved via characteristic equations.

IntuitionIntuition: oscillation and the superposition principle

A mass on a spring, a swinging pendulum with small amplitude, and the current in an RLC circuit all obey the same shape of law: an unknown function together with its derivatives combine linearly. If you know two separate motions that each satisfy the law, their sum (or any weighted combination) also satisfies it — this is the superposition principle. It is the reason we can build every possible motion out of just two building-block solutions.

Damped harmonic oscillator

y'' + 2\zeta y' + y = 0

: the solution oscillates inside the decaying exponential envelope

\pm e^{-\zeta t}

SchoolPrecise statement: second-order linear equations with constant coefficients

Definition: Homogeneous linear differential equation

A second-order equation of the form $ay'' + by' + cy = 0$ , where $a,b,c$ are constants and $a\neq 0$ , is called homogeneous linear because the right-hand side is zero and $y$ appears only to the first power in every term.

ay'' + by' + cy = 0

Because differentiating $e^{rx}$ just multiplies it by $r$ , we try the ansatz $y=e^{rx}$ . Substituting into the equation and dividing by the common factor $e^{rx}\neq 0$ turns the differential equation into a plain quadratic equation in $r$ :

ar^2+br+c=0

The three cases of the characteristic equation $ar^2+br+c=0$ and the resulting general solution
Discriminant	Roots	General solution
$\Delta>0$	Two distinct real roots $r_1\neq r_2$	$y = c_1 e^{r_1 x} + c_2 e^{r_2 x}$
$\Delta=0$	One repeated real root $r$	$y = (c_1 + c_2 x) e^{r x}$
$\Delta<0$	Complex conjugate roots $r = \alpha \pm i\beta$	$y = e^{\alpha x}\left(c_1 \cos(\beta x) + c_2 \sin(\beta x)\right)$

UndergraduateUniversity level: theorems and proofs

Theorem: Superposition principle

If $y_1$ and $y_2$ both satisfy $ay'' + by' + cy = 0$ , then for any constants $c_1,c_2$ the combination $y=c_1y_1+c_2y_2$ also satisfies the equation. If moreover the Wronskian $W(y_1,y_2) = y_1y_2' - y_2y_1'$ is nonzero at some point, every solution of the equation has this form.

Why is it true?

Differentiation is a linear operation: the derivative of a sum is the sum of derivatives, and constants factor out. Since the left side of the equation is built only from $y$ , $y'$ , $y''$ multiplied by constants and added together, plugging in a combination of two solutions just adds up two copies of zero.

Proof

Suppose $L[y] = ay''+by'+cy$ . Since differentiation satisfies $(u+v)'=u'+v'$ and $(ku)'=ku'$ for a constant $k$ , the operator $L$ is linear: $L[c_1y_1+c_2y_2] = a(c_1y_1+c_2y_2)'' + b(c_1y_1+c_2y_2)' + c(c_1y_1+c_2y_2) = c_1(ay_1''+by_1'+cy_1) + c_2(ay_2''+by_2'+cy_2) = c_1L[y_1]+c_2L[y_2]$ .

Because $y_1,y_2$ solve the equation, $L[y_1]=0$ and $L[y_2]=0$ , so $L[c_1y_1+c_2y_2]=c_1\cdot 0+c_2\cdot 0=0$ : the combination is again a solution. This proves the first half.

For the second half, the theory of linear ODEs guarantees that the solution space of a second-order equation is exactly two-dimensional (an initial condition $y(x_0)=y_0$ , $y'(x_0)=y_0'$ picks out a unique solution, giving two free parameters). A nonzero Wronskian $W(y_1,y_2)=y_1y_2'-y_2y_1'$ at some point $x_0$ means the linear system $c_1y_1(x_0)+c_2y_2(x_0)=y_0$ , $c_1y_1'(x_0)+c_2y_2'(x_0)=y_0'$ has a unique solution $(c_1,c_2)$ for every choice of initial data. So every solution matches some $c_1y_1+c_2y_2$ for suitable constants, which means $\{y_1,y_2\}$ spans the whole two-dimensional solution space.

Theorem: Structure of the general solution from the characteristic roots

The general solution of $ay'' + by' + cy = 0$ is determined entirely by the roots of the characteristic equation $ar^2+br+c=0$ : distinct real roots give $y = c_1 e^{r_1 x} + c_2 e^{r_2 x}$ , a repeated real root gives $y = (c_1 + c_2 x) e^{r x}$ , and complex conjugate roots $r = \alpha \pm i\beta$ give $y = e^{\alpha x}\left(c_1 \cos(\beta x) + c_2 \sin(\beta x)\right)$ .

Why is it true?

Exponentials are the natural building blocks because differentiating $e^{rx}$ just rescales it by $r$ ; feeding this ansatz into a linear equation with constant coefficients collapses calculus into algebra — finding roots of a polynomial.

Proof

Try the ansatz $y=e^{rx}$ . Then $y'=re^{rx}$ and $y''=r^2e^{rx}$ ; substituting into $ay'' + by' + cy = 0$ gives $(ar^2+br+c)e^{rx}=0$ . Since $e^{rx}$ is never zero, this forces the characteristic equation $ar^2+br+c=0$ . By the quadratic formula, $r = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$ .

Case $\Delta>0$ : the two roots $r_1\neq r_2$ give solutions $e^{r_1x}$ and $e^{r_2x}$ . Their Wronskian is $(r_2-r_1)e^{(r_1+r_2)x}\neq 0$ , so by the superposition theorem the general solution is $y = c_1 e^{r_1 x} + c_2 e^{r_2 x}$ .

Case $\Delta=0$ : there is one repeated root $r=-b/(2a)$ , giving only one exponential solution $e^{rx}$ . To find a second independent solution, try $y_2=v(x)e^{rx}$ (reduction of order) and substitute into the equation; because $r$ is a double root of $ar^2+br+c$ , the terms in $v$ and $v'$ cancel, leaving $av''e^{rx}=0$ , i.e. $v''=0$ , so $v=c_1+c_2x$ . This yields the second solution $xe^{rx}$ , and the general solution $y = (c_1 + c_2 x) e^{r x}$ .

Case $\Delta<0$ : the roots are complex conjugates $r = \alpha \pm i\beta$ with $\alpha=-b/(2a)$ , $\beta=\sqrt{-\Delta}/(2a)$ . Euler's formula $e^{i\beta x}=\cos(\beta x)+i\sin(\beta x)$ turns the complex solutions $e^{(\alpha\pm i\beta)x}$ into two independent real solutions $e^{\alpha x}\cos(\beta x)$ and $e^{\alpha x}\sin(\beta x)$ (by taking the real and imaginary parts, which are themselves solutions since $L$ has real coefficients), giving the general solution $y = e^{\alpha x}\left(c_1 \cos(\beta x) + c_2 \sin(\beta x)\right)$ .

UndergraduateReal-World Applications and Worked Examples

The three cases of the characteristic equation are not just algebra — they describe three physically distinct behaviours of a damped spring-mass system $my'' + cy' + ky = 0$ (mass $m$ , damping $c$ , stiffness $k$ ): underdamped oscillation that dies out while wiggling ( $\Delta<0$ ), overdamped return with no wiggle at all ( $\Delta>0$ ), and critically damped return, the fastest possible non-oscillating return ( $\Delta=0$ ) — exactly the design goal of a car's shock absorber or a door closer.

Example: Finding a general solution: distinct real roots

Solve $y'' - 5y' + 6y = 0$ .

Solution

The characteristic equation is $r^2-5r+6=0$ .

Factoring, $(r-2)(r-3)=0$ , so the roots are $r_1=2$ and $r_2=3$ : two distinct real roots.

By the theorem above (case $\Delta>0$ ), the general solution is $y = c_1e^{2x}+c_2e^{3x}$ , where $c_1,c_2$ are arbitrary constants fixed by initial conditions.

Example: Undetermined coefficients for a forced equation

Find a particular solution of $y'' - y = e^{2x}$ .

Solution

The homogeneous equation $y''-y=0$ has characteristic equation $r^2-1=0$ , with roots $r=\pm 1$ ; the forcing term $e^{2x}$ has exponent $2$ , which is not one of these roots, so there is no resonance.

Try the ansatz $y_p = Ae^{2x}$ . Then $y_p''=4Ae^{2x}$ , and substituting gives $4Ae^{2x}-Ae^{2x}=3Ae^{2x}$ , which must equal $e^{2x}$ .

Matching coefficients, $3A=1$ , so $A=\tfrac{1}{3}$ , giving the particular solution $y_p=\tfrac{1}{3}e^{2x}$ . The full general solution is $y=c_1e^{x}+c_2e^{-x}+\tfrac{1}{3}e^{2x}$ .

What is the general solution of $y''-7y'+12y=0$ ?

For $y''+2y'+5y=0$ , what kind of roots does the characteristic equation have, and what is the general solution?

A car's shock absorber, modeled by $my'' + cy' + ky = 0$ , is designed to return the car body to rest as fast as possible without any bouncing. Which case of the characteristic equation should the engineer aim for?

To find a particular solution of $y''-y=e^{2x}$ by undetermined coefficients, which trial form should you use?

References

William E. Boyce, Richard C. DiPrima (2017). Elementary Differential Equations and Boundary Value Problems
Lawrence Perko (2001). Differential Equations and Dynamical Systems