Differential Equations Study Notes

Introduction to Differential Equations

• Definition of Differential Equations (D.E.): An equation that contains the derivative of an unknown function, which is the dependent variable.

  • Example 1: A first-order differential equation can be expressed as $y'(t)$ where:

  • $t$: independent variable

  • $y$: dependent variable

  • Example 2: A second-order differential equation can be given as $y''(t) + 2y(t) + heta(t) = 0$.

  • In this case, $y(t)$ represents the unknown function, while the terms include its second-order derivative and other functions.

  • Newton's Second Law: Expressed as $F = ma$, rephrased to $v'(t) = x''(t) = - heta(t) = m x''(t)$.

  • Variables:

    • $x(t)$: position

    • $u(t)$: velocity

    • $a(t)$: acceleration

Solving Differential Equations

• Methodology to Solve D.E.:

  1. Use the provided function to compute necessary derivatives.

  2. Substitute back into the equation and simplify.

  3. Confirm that the left-hand side (L.H.S) equals the right-hand side (R.H.S).

• Example 3: Verify whether $y(t) = e^{t} - Y'(t) e^{t} + zy = 2e^{2} + 2e^{t} - 4e^{t}$ is a solution to the differential equation defined as:

  • Results in $Y''_{4}Y = 0$ where R.H.S = 0.

Classification of Differential Equations

• Ordinary Differential Equations (ODE): The unknown function depend on only one independent variable.

  • Example: $f(t)$, often referred to as a simple ODE.

• Partial Differential Equations (PDE): The unknown function depends on more than one independent variable.

  • Example: $f(t, s)$ is a PDE where partial derivatives exist.

• Continuing Classification:

  • The order of the D.E. is determined by the highest degree derivative present in the equation.

  • Linear vs Non-linear:

  • Linear D.E.: All terms involving the unknown function and its derivatives occur to the first power, with coefficients dependent only on the independent variable.

  • Non-linear D.E.: The terms violate linearity conditions (i.e., powers greater than 1).

General Forms of Differential Equations

• General Form of Linear ODE:
  $a<em>{n}(x) y^{(n)} + a</em>{n-1}(x) y^{(n-1)} + … + a<em>1(x) y' + a</em>0(x) y = F(x)$

  • Where:

  • $y^{(n)}$: nth derivative of y

  • $F(x)$: a function of the independent variable

• Determining Linear vs Non-linear:

  • Example a: $y'(t) + y^2 e^0$ is non-linear.

  • Example b: $f(x) + x f'(x) = 2$ is linear.

• Explicit vs Implicit Solution:

  • Explicit: $y = g(x)$ that satisfies the differential equation for all $x$ in the interval considered.

  • Implicit: $F(x, y, y', …, y^{(n)}) = 0$ involves mixing the dependent variable with its derivatives, proving more complex to isolate solutions.

Examples of Solving D.E.

• Example: Verify if $y = 3 ext{sin}(2x) + e^{-x}$ is a solution:

  • Compute derivatives:
    $y' = 6 ext{cos}(2x) - e^{-x}$
    $y'' = -12 ext{sin}(2x) - e^{-x}$

  • Check against the defined D.E.

• Example 2: Determine whether $heta(x) = e^{x} - x$ is a solution to the equation relating to $L.H.S = R.H.S$.

Initial Value Problems (IVP)

• Definition: An initial value problem is a D.E. that includes additional initial conditions, such as $y(0) = y_0$.

• Example of IVP: Solve for $y$ given that the general solution is $y(x) = C<em>1 e^x + C</em>2 e^{-2x}$. Determine constants using conditions like $y(0) = 2$ and $y'(0) = 1$.

Direction Fields and Graphical Approach

• Definition of Direction Field: A visual representation that indicates the slope of solutions to a D.E. across a certain plane.

• Example: For $dy/dx = x - 4$, the direction at any point $(x,y)$ on the plane can be plotted to visualize the behavior of the solution curves.

  • Use slope strategies to sketch and utilize for initial value problems.

Autonomous ODE

• Definition: An autonomous ODE is one that is written as $dy/dx = f(y)$ and does not contain the independent variable explicitly.

• Example: In the population model $dP/dt = -2P(P - 4)$, where $P$ changes with time $t$, it indicates how population dynamics can be depicted without directly depending on time.