Unit 5: Introduction to Differential Equations

Introduction and Definitions

A differential equation (DE) is an equation that involves an unknown function and one or more of its derivatives.

Differential equations are used to model real-world phenomena where a quantity's rate of change depends on the quantity itself (e.g., population growth, radioactive decay, circuits).

• Ordinary Differential Equation (ODE): Contains derivatives with respect to only one independent variable. (e.g., `dy/dx`, `d²y/dt²`). This unit focuses *only* on ODEs.
• Partial Differential Equation (PDE): Contains partial derivatives with respect to two or more independent variables. (e.g., `∂u/∂t`, `∂²u/∂x²`).

Order and Degree of a DE

Order

The order of a differential equation is the order of the highest-order derivative present in the equation.

Degree

The degree of a differential equation is the power (exponent) of the highest-order derivative, *after* the equation has been cleared of any fractions or radicals in its derivatives.

Examples:

1. (d²y/dx²)³ + 5(dy/dx)⁷ + y = 0
   • Highest derivative is `d²y/dx²`, so Order = 2.
   • The power of this highest derivative is 3, so Degree = 3.
2. d³y/dx³ + 2x = √(1 + dy/dx)
   • Highest derivative is `d³y/dx³`, so Order = 3.
   • The power of this derivative is 1, so Degree = 1. (The radical on `dy/dx` is irrelevant as it's a lower-order term).
3. y'' = (1 + (y')²)³ᐟ²
   • First, clear the radical/fractional power: Square both sides.
   • (y'')² = (1 + (y')²)³
   • Highest derivative is `y''`, so Order = 2.
   • The power of `y''` is 2, so Degree = 2.

General and Particular Solutions

General Solution

A general solution to a DE is a solution that contains one or more arbitrary constants (like `C`, `A`, `B`). The number of arbitrary constants is equal to the order of the DE.

Example: For `dy/dx = 2x`, the general solution is `y = x² + C`.

Particular Solution

A particular solution is a solution obtained from the general solution by assigning specific values to the arbitrary constants. This is usually done using "initial conditions" or "boundary conditions."

Example: For `y = x² + C`, if we are given the initial condition `y(0) = 5`, we can find `C`:

`5 = (0)² + C` ⇒ `C = 5`.
The particular solution is `y = x² + 5`.

Formation of a Differential Equation

This is the reverse process: given a general solution (a family of curves), find the DE they satisfy by eliminating the arbitrary constants.

Method:

1. Count the number of arbitrary constants (say, `n`). The order of your final DE will be `n`.
2. Differentiate the given equation `n` times.
3. You now have `n+1` equations (the original plus the `n` derivatives).
4. Use algebra (substitution, elimination) to eliminate all `n` constants from these equations.

Example: Form the DE for the family of curves `y = A cos(x) + B sin(x)` (n=2 constants: A, B)

1. Original: `y = A cos(x) + B sin(x)` --- (1)
2. Differentiate 1: `y' = -A sin(x) + B cos(x)` --- (2)
3. Differentiate 2: `y'' = -A cos(x) - B sin(x)` --- (3)
4. Eliminate: Notice that equation (3) is the negative of equation (1).
   `y'' = -(A cos(x) + B sin(x))`
   Substitute (1): `y'' = -y`

The resulting DE is y'' + y = 0 or d²y/dx² + y = 0.

Solving DEs: Variable Separable

This is the simplest method for solving first-order, first-degree DEs.

A DE is variable separable if it can be algebraically rearranged into the form:
f(y) dy = g(x) dx
(All `y` terms on one side, all `x` terms on the other).

Solution: Integrate both sides and add a constant of integration `C`.

∫ f(y) dy = ∫ g(x) dx + C

Example: Solve `dy/dx = xy²`

1. Separate variables: `dy / y² = x dx`
2. Integrate both sides: `∫ y⁻² dy = ∫ x dx + C`
3. `-y⁻¹ = (x²/2) + C`
4. -1/y = x²/2 + C (This is the general solution).

Solving DEs: Homogeneous Equation

A first-order DE dy/dx = F(x, y) is homogeneous if `F(x, y)` is a homogeneous function of degree 0.
This means `F(tx, ty) = F(x, y)`, or more simply, it can be written as a function of `y/x`.
Example: `dy/dx = (x² + y²) / (xy)`

Solution Method:

1. Use the substitution: y = vx
2. Differentiate with respect to `x` (using product rule): dy/dx = v(1) + x(dv/dx)
3. Substitute both `y = vx` and `dy/dx = v + x(dv/dx)` into the original DE.
4. The `x` variables will cancel, leaving a new DE involving only `v` and `x`.
5. This new DE will always be variable separable. Solve it for `v`.
6. Substitute v = y/x back in to get the final solution.

Solving DEs: Linear Differential Equation

This is one of the most important types of DEs.

A first-order DE is linear if it can be written in the standard form:
dy/dx + P(x)y = Q(x)
where P(x) and Q(x) are functions of `x` only (or constants).

Solution Method (3 Steps):

1. Step 1: Identify `P(x)` and `Q(x)` from the standard form.
   (Make sure the coefficient of `dy/dx` is 1).
2. Step 2: Find the Integrating Factor (I.F.).
   I.F. = e^(∫ P(x) dx)
3. Step 3: The general solution is given by the formula:
   y · (I.F.) = ∫ [ Q(x) · (I.F.) ] dx + C