Section 19.5
Substitutions
When standard methods fail, a clever change of variables can transform the impossible into the solvable.
1
Introduction
Just like U-Substitution in integration, substitution in DEs is about simplifying structure.
The most common type is the Homogeneous Differential Equation, where the derivative only depends on the ratio .
2
Homogeneous Equations
Definition
A DE is homogeneous if it can be written as .
Recipe
- Substitute: Let , which means .
- Differentiate: Product rule gives .
- Replace: .
- Separate: .
- Solve & Back-Sub: Find , then replace with .
3
Visual: Radial Symmetry
Interactive: Slope depends on angle
For homogeneous equations, the slope is constant along any line through the origin. This symmetry is why substitution works.
4
Worked Examples
Example 1: Homogeneous
Solve .
1. Standard Form:
.
2. Substitute:
. .
3. Separate:
.
.
4. Integrate:
.
5. Back-Sub:
.
.
Example 2: Another Homogeneous
Solve .
1. Standard Form:
.
2. Substitute:
.
.
.
3. Integrate:
.
(absorbing constant).
4. Final:
.
.
Example 3: Linear Coefficients substitution
Solve .
1. Substitute:
Let . Then , so .
2. Transform:
.
3. Separate:
.
.
.
4. Back-Sub:
.
.
5
Practice Quiz
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