What is a differential equation?
An equation that relates a function with its derivatives.
What is a general solution to a differential equation?
A solution that contains arbitrary constants, representing a family of solutions.
What does it mean to verify a solution to a differential equation?
To confirm that a given function satisfies the differential equation when substituted into it.
Steps to verify if $y=f(x)$ is a solution to $\frac{dy}{dx} = g(x)$?
1. Find $\frac{dy}{dx}$. 2. Substitute $y=f(x)$ and $\frac{dy}{dx}$ into the differential equation. 3. Verify that the equation holds true.
How do you verify a solution involving $e^{kx}$?
1. Differentiate $y = e^{kx}$ to get $\frac{dy}{dx} = ke^{kx}$. 2. Substitute both into the differential equation. 3. Check for equality.
How do you verify a solution involving trigonometric functions?
1. Differentiate $y=sin(kx)$ or $y=cos(kx)$ using appropriate rules. 2. Substitute $y$ and $\frac{dy}{dx}$ into the differential equation. 3. Simplify and check if the equation holds.
Explain the process of verifying a solution to a differential equation.
Differentiate the proposed solution, then substitute the derivative and the original solution into the differential equation. Check if the equation holds true.
Why can a differential equation have infinitely many solutions?
Differential equations often have general solutions that include arbitrary constants. Varying these constants produces different, but valid, solutions.
What is the role of derivatives in verifying solutions to differential equations?
Derivatives are used to transform the proposed solution into a form that can be directly compared with the differential equation.
