Solutions and Initial Conditions
A solution to a DE is a function that satisfies the equation when substituted in. But there's usually a whole family of solutions - you need initial conditions to pick the specific one.
Types of Solutions
General vs Particular Solution
Medium
Verify y = Ce^(2x) is a solution to y' = 2y, then find the particular solution with y(0) = 3.
1
Verify it's a solution
y = Ce^(2x)
y' = 2Ce^(2x) = 2y ✓
2
Apply initial condition
y(0) = 3
Ce^(0) = 3
C = 3
3
Write particular solution
y = 3e^(2x)
Answer:
Particular solution: y = 3e^(2x)
How Many Initial Conditions?
For an nth-order ODE, you need n initial conditions to get a unique solution.
• First-order: need y(x₀) = y₀
• Second-order: need y(x₀) = y₀ AND y'(x₀) = y₁
Key Takeaways
General solution has arbitrary constants
Particular solution uses initial conditions to find constants
nth-order ODE needs n initial conditions for unique solution