Differential Equations 1

Posted on February 2, 2019
Tags: physics

1 Intro

graph TD A[Identify the Differential Equation] -->|"Is the equation in the form dy/dx = f(x)"| B[Linear] A -->|"Is the equation in the form dy/dx = f(x, y)"| C[Nonlinear] B -->|Are the coefficients of y and its derivatives only functions of x?| D[Linear] B -->|Otherwise| E[Nonlinear] C -->|Is the equation separable or homogeneous?| F[Linear] C -->|Otherwise| G[Nonlinear] F -->|Linear terms or linear combinations in numerator and denominator?| H[Linear] F -->|Otherwise| I[Nonlinear] H -->|Is it a first-order linear ODE?| J[Linear] H -->|Otherwise| K[Nonlinear]

2 Verifying potential solution differential eqn

Differential equation is like a specificiation. Many potential solutions(equations) can fit that spec.
You must check the solution fits the spec.

Plug in the equation \(y = \frac{1}{16}x^4\) in the differential equation.
Result: \(\frac{d}{dx}(\frac{1}{16}x^4) ?= x(\frac{1}{16}x^4)^{0.5}\)
Check if the equality is satisfied

2.0.1 Differential Equations

Properties

  • Order: order of highest derivative
  • (Non)Linear:
  • (ODE)PDE
  • Homogeneity
2.0.1.0.1 Initial Value Problems

notation: \(y' := \frac{dy}{dx}\) \(y := y(x)\)

Given Differential Equation \[y'' - 5y' + 6y = 0\]
Solve for set of equations that satisfy Differential equation:

DSolve[y''[x] - 5*y'[x] + 6 y[x] == 0, y[x], x]

\[\left\{\left\{y(x)\to c_1 e^{2 x}+c_2 e^{3 x}\right\}\right\}\]

Given Initial conditions: \[y(0)=3\] \[y'(0)=1\] Solve by finding the specific functions that satisfy the initial condition and the given differential equation

  • \(y(0)=3\)
    • Plug into previous solution \(y(x)=c_1 e^{2 x}+c_2 e^{3 x}\)
    • \(y(0)=c_1 +c_2 = 3\)
  • \(y'(0)=2\)
    • Find derivative of previous solution \(y(x)=c_1 e^{2 x}+c_2 e^{3 x}\)
      • \(y'(x) = 2c_1 + 3c_2\)
    • \(y'(0)=2c_1 + 3c_2 = 1\)

Solve system of linear equations for constants \(c_1\),\(c_2\)

\[ \begin{aligned} c_1 + c_2 &= 3\\ 2c_1 + 3c_2 &= 1 \end{aligned} \quad\Longleftrightarrow\quad \begin{bmatrix} 1 & 1 \\ 2 & 3 \\ \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ \end{bmatrix}= \begin{bmatrix} \text{3} \\ \text{1} \\ \end{bmatrix} \]

mat = {{1, 1}, {2, 3}}
vars = {{c1}, {c2}}
b = {{3}, {1}}

Solve[{mat . vars == b}, {c1, c2}]

\[\{\{c_1\to 8,c_2\to -5\}\}\]

Solution: \[y = 8e^{2x}-5e^{3x}\]

2.0.2 Examples

2.0.2.0.1 Radioactive Decay , substance amount y[t]

\[\frac{dy}{dt} = -ky(t)\]

DSolve[y'[t] == -k*y[t], y[t], t] // TeXForm

\[\left\{\left\{y(t)\to c_1 e^{-k t}\right\}\right\}\]

\[y(0)=k\]

2.0.2.0.2 Uniqueness and Existence Problems

Theorem:
Given f(x,y) continous