Theory of Differential Equations

The power the differential is raised to.

The dependent variable and its derivatives are all not non-linear. \[\begin{aligned} \underbrace{\frac{d^2 y}{d t}} &\quad \underbrace{\cos(x) \frac{dy}{dx}} &\quad \underbrace{\frac{dy}{dt} \frac{d^3 y}{dt^3}} &\quad \underbrace{y’ = e^y} &\quad \underbrace{y \frac{dy}{dx}} \\ \text{linear} &\quad \text{linear} &\quad \text{non-linear} &\quad \text{non-linear} &\quad \text{non-linear} \end{aligned}\]

Independent variable does not appear in the equation.

Independent variable does appear in the equation.

Our initial guess for the form of a solution, e.g. \(y_p = A \cos(t) + B \sin(t)\).

A quadratic equation that pops out during the application of the Frobenius method.

A function is analytic at a point if it can be expressed as a convergent power series in a neighborhood of that point.

When \(p(x)\) and \(q(x)\) are analytic at that point.

If \(P(x) = (x-x_0)p(x)\) and \(Q(x) = (x-x_0)^2 q(x)\) are both analytic at \(x_0\).

Not regular.

A sequence of functions \(f_n\) converges in mean to \(f\) on \([a,b]\) if \(\lim_{n \to \infty} \int^b_a |f_n(x) - f(x)|^2 \, dx = 0\).

A sequence of functions \(f_n\) converges pointwise to \(f\) on \([a,b]\) if \(\lim_{n \to \infty} f_n(x) = f(x)\) for every \(x \in [a,b]\).

A sequence of functions \(f_n\) converges uniformly to \(f\) on \([a,b]\) if \(\lim_{n \to \infty} \sup_{x \in [a,b]} |f_n(x) - f(x)| = 0\).

A point where the derivative of the dependent variable with respect to the independent variable is zero.

Trajectories approach the equilibrium point from all directions and eigenvalues are real and negative.

Trajectories move away from the equilibrium point in all directions and eigenvalues are real and positive.

Trajectories orbit around the equilibrium point with eigenvalues that are purely imaginary.

Trajectories approach the equilibrium point in one direction and move away in another, with eigenvalues having opposite signs.

Trajectories spiral away from the equilibrium point with eigenvalues having positive real parts and non-zero imaginary parts.

\[\frac{dy}{dx} = f(x, y)\]

\[\frac{dy}{dx} = f(x) g(y) \implies \int \frac{dy}{g(y)} = \int f(x) \, dx\]

\[\frac{dy}{dx} = f\left(\frac{y}{x}\right)\] with substitution: \(y(x) = x v(x)\).

\[\frac{dy}{dx} + p(x) y = q(x)\]

Note, the coefficient of \(y’(x)\) must be 1. \[\phi(x) = \exp\left(\int p(x) \, dx\right)\] Multiplying the Linear Standard Form with \(\phi(x)\) yields: \[\frac{d}{dx}(\phi y) = \phi(x) q(x) \implies y = \phi^{-1} \int \phi q(x) \, dx\]

A first-order ODE is exact if it can be written in the form: \[M(x,y) \, dx + N(x,y) \, dy = 0\] where \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\). The solution is then given by: \(F(x,y) = C\) where \(F(x,y)\) satisfies \(\frac{\partial F}{\partial x} = M(x,y)\) and \(\frac{\partial F}{\partial y} = N(x,y)\).

\[y’’ + p(x)y’ + q(x)y = r(x)\]

\[\frac{d^2 y}{dx^2} + f\left(y, \frac{dy}{dx}\right) = 0\] is reducible to the first-order ODE \[p \frac{dp}{dy} + f(y, p) = 0\] with substitution \(p = \frac{dy}{dx}\).

When \(p(x)\) and \(q(x)\) are constants: \[y’’ + a_1 y’ + a_0 y = 0\]

Solve the characteristic equation: \[\lambda^2 + a_1 \lambda + a_0 = 0\] Cases:

• \(\lambda_1, \lambda_2\) are real and distinct
• \(\lambda_1, \lambda_2\) are real and coincide (same)
• \(\lambda_1, \lambda_2\) are complex conjugates

In each case, the solution of \(y(x)\) becomes:

• \(y(x) = C e^{\lambda_1 x} + D e^{\lambda_2 x}\)
• \(y(x) = C e^{\lambda_1 x} + D x e^{\lambda_1 x}\)
• \(y(x) = e^{\alpha x}(A \cos(\beta x) + B \sin(\beta x))\) by DeMoivre’s Theorem

\[y(x) = y_h(x) + y_p(x)\] Guesses for \(y_p(x)\):

• For \(r(x) = P_n(x)\) (polynomial of degree \(n\)), try \(y_p(x) = Q_n(x)\)
• For \(r(x) = e^{\alpha x}\), try \(y_p(x) = A e^{\alpha x}\)
• For \(r(x) = \sin(\beta x)\) or \(r(x) = \cos(\beta x)\), try \(y_p(x) = A \sin(\beta x) + B \cos(\beta x)\)
• For products of the above forms, try products of the corresponding forms
• If \(y_p(x)\) is already a solution of the homogeneous equation, multiply by \(x\) or \(x^k\) until linearly independent

Note, that we embark on this approach because the second order standard form is not solvable in general with elementary functions!

Pick ansatz of the form \[y = \sum^{\infty}_{n=0} a_n z^n\] and take derivatives as required. For example: \[\frac{dy}{dz} = \sum^{\infty}_{n=1} n a_n z^{n-1}, \quad \frac{d^2 y}{dz^2} = \sum^{\infty}_{n=2} n(n-1) a_n z^{n-2}\] and substitute them into the ODE. Then solve by rearranging indices as necessary to obtain a recurrence relation. Apply the initial conditions and then guess the closed-form solution of the recurrence relation. Change back to the original variables if required.

If \(x_0\) is an ordinary point of the differential equation \[y’’ + p(x)y’ + q(x)y = 0\] then the general solution in a neighbourhood \(|x - x_0| < R\) may be represented as a power series.

\[r(r-1) + p_0 r + q_0 = 0\]

For an \(n^{\text{th}}\) order linear ODE with variable coefficients: \[a_n(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_1(x) y’ + a_0(x) y = f(x)\]

We assume a solution of the form: \[y(x) = \sum^{\infty}_{k=0} c_k (x-x_0)^k\]

Taking derivatives and substituting yields a recurrence relation for coefficients \(c_k\), typically allowing us to determine \(c_n\) in terms of \(c_0, c_1, \dots, c_{n-1}\).

Any \(n^{\text{th}}\) order ODE can be formulated as a system of \(n\) first order ODEs.

For \(y^{(n)} = f(x, y, y’, \dots, y^{(n-1)})\), set \(y_i = y^{(i-1)}\) for \(i = 1,2,\dots,n\) to obtain: \[y_i’ = y_{i+1} \text{ for } i = 1,2,\dots,n-1\] \[y_n’ = f(x, y_1, y_2, \dots, y_n)\]

\[A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D\frac{\partial u}{\partial x} + E\frac{\partial u}{\partial y} + F u = 0\]

• Parabolic equation: \(B^2 - 4AC = 0\) (Heat Equation)
• Hyperbolic equation: \(B^2 - 4AC > 0\) (Wave Equation)
• Elliptic equation: \(B^2 - 4AC < 0\) (Laplace Equation)

\[U(x,y) = X(x) Y(y)\] then \(U_x = Y X’\) and \(U_y = Y’ X\). Rewrite the PDE with these substitutions, then divide through by \(XY\). Integrate and solve.

When a PDE is difficult to solve directly, changing variables can transform it into a simpler form.

For a second-order PDE, the transformation \(u = u(\xi, \eta)\) where \(\xi = \xi(x,y)\) and \(\eta = \eta(x,y)\) requires computing: \[\frac{\partial u}{\partial x} = \frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial x} + \frac{\partial u}{\partial \eta}\frac{\partial \eta}{\partial x}\] \[\frac{\partial u}{\partial y} = \frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial y} + \frac{\partial u}{\partial \eta}\frac{\partial \eta}{\partial y}\]

And similarly for second-order derivatives. The canonical transformations are:

• For hyperbolic: \(\xi = x + y, \eta = x - y\) (characteristic coordinates)
• For parabolic: \(\xi = x, \eta = y - f(x)\) (transformation along characteristics)
• For elliptic: \(\xi = x + iy, \eta = x - iy\) (complex characteristics)

• \(\lambda_2 < \lambda_1 < 0 \implies\) stable node
• \(0 < \lambda_1 < \lambda_2 \implies\) unstable node
• \(\lambda_1 = \lambda_2, \lambda_1 > 0 \implies\) unstable star
• \(\lambda_1 = \lambda_2, \lambda_1 < 0 \implies\) stable star
• \(\lambda_1 < 0 < \lambda_2 \implies\) unstable saddle node
• \(\operatorname{Re}(\lambda_1) = 0 \implies\) centre, stable
• \(\operatorname{Re}(\lambda_1) < 0 \implies\) stable focus
• \(\operatorname{Re}(\lambda_1) > 0 \implies\) unstable focus

For a linear system \(\dot{\mathbf{x}} = \mathbf{A} \mathbf{x}\), the real canonical form depends on the eigenvalues of \(\mathbf{A}\):

• Real distinct eigenvalues \(\lambda_1 \neq \lambda_2\):

\[\mathbf{A}_{\text{canonical}} = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix}\]

• Real repeated eigenvalues \(\lambda_1 = \lambda_2\) with linearly independent eigenvectors:

\[\mathbf{A}_{\text{canonical}} = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_1 \end{pmatrix}\]

• Real repeated eigenvalues \(\lambda_1 = \lambda_2\) with one linearly independent eigenvector:

\[\mathbf{A}_{\text{canonical}} = \begin{pmatrix} \lambda_1 & 1 \\ 0 & \lambda_1 \end{pmatrix}\]

• Complex conjugate eigenvalues \(\lambda = \alpha \pm i\beta\):

\[\mathbf{A}_{\text{canonical}} = \begin{pmatrix} \alpha & \beta \\ -\beta & \alpha \end{pmatrix}\]

\[W(f_1, f_2, \dots, f_n)(x) = \begin{vmatrix} f_1(x) & f_2(x) & \dots & f_n(x) \\ f_1’(x) & f_2’(x) & \dots & f_n’(x) \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \dots & f_n^{(n-1)}(x) \end{vmatrix}\] Note that if a set of functions is linearly dependent, then its Wronskian will equal 0.

A set of functions \(\{\phi_n\}_{n=1,2,3,\dots}\) is said to be orthogonal on the interval \([a,b]\) with respect to the inner product defined by \[(f, g)_w = \int^b_a w(x)f(x)g(x) \, dx\] with weight function \(w(x) > 0\), if \((\phi_n,\phi_m)_w = 0\) for \(m \neq n\).

A set \(\{\phi_n\}_{n=1,2,3,\dots}\) is orthonormal when in addition to being orthogonal, \((\phi_n,\phi_n) = 1\), for \(n = 1,2,3,\dots\).

\[x^2 y’’ + a_1 x y’ + a_0 y = 0\] You can solve this by either letting \(x = e^t\) or using the ansatz \(y = x^\lambda\). The characteristic equation is \(\lambda^2 + (a_1 - 1) \lambda + a_0 = 0\). If you are blessed with the inhomogeneous case of above, just use method of undetermined coefficients.

\[(1 - x^2)y’’ - 2x y’ + n(n+1)y = 0\] Solutions are Legendre polynomials.

\[x^2 y’’ + x y’ + (x^2 - \nu^2) y = 0\]

Bessel function of the first kind of order \(\alpha\): \[J_\alpha(x) = \sum^{\infty}_{m=0} \frac{(-1)^m}{\Gamma(m+1)\Gamma(m+\alpha+1)} \left(\frac{x}{2}\right)^{2m+\alpha}\] implies \[\frac{d}{dx} \left[x^\alpha J_\alpha(x)\right] = x^\alpha J_{\alpha-1}(x)\] implies \[\int^r_0 x^n J_{n-1}(x) \, dx = r^n J_n( r) \text{ for } n = 1, 2, 3, \dots\]

The DE admits solutions:

• Case 1: \(2\nu \notin \mathbb{Z}\): \(y(x) = A J_{\nu}(x) + B J_{-\nu}(x)\), \(J_{\nu}(x)\), \(J_{-\nu}(x)\) linearly independent
• Case 2: \(2\nu \in \mathbb{Z}\): \(y(x) = A J_{\nu}(x) + B J_{-\nu}(x)\)
• Case 3: \(\nu \in \mathbb{Z}\): \(J_{\nu}(x)\), \(J_{-\nu}(x)\) linearly dependent, \(y(x) = A J_{\nu}(x) + B Y_{\nu}(x)\)

\[x y’’ + (1-x)y’ + n y = 0\]

\[y’’ - 2 x y’ + 2 n y = 0\]

\[(p y’)’ + (q + \lambda r) y = 0\] Note that Bessel, Laguerre, Hermite and Legendre equations can all be written in this form. Furthermore, any 2nd order linear homogeneous ODE \(y’’ + a_1(x)y’ + [a_2(x) + \lambda a_3(x)]y = 0\) may be written in this form.

\[\frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}\]

\[\frac{\partial^2 u}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 u}{\partial t^2}\]

\[\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\]

\[y(x) = \frac{a_0}{2} + \sum^{N}_{n=1} (a_n \cos(n x) + b_n \sin(n x))\] \[a_n = \frac{1}{\pi} \int^{\pi}_{-\pi} y(x) \cos(nx) \, dx, \quad n = 0, 1, 2, \dots\] \[b_n = \frac{1}{\pi} \int^{\pi}_{-\pi} y(x) \sin(nx) \, dx, \quad n = 1, 2, \dots\]

\[\frac{\|f\|^2}{L} = \frac{1}{L} \int^{L}_{-L} f^2 \, dx = \frac{a_0}{2} + \sum^{\infty}_{n=1} (a_n^2 + b_n^2)\]