01 Key Concepts
What is a Differential Equation?
An equation involving a function and one or more of its derivatives, such as dy/dx = 3y.
Order of a Differential Equation
Determined by the highest derivative appearing in the equation; dy/dx=3y is first-order, while equations with y'' are second-order.
Separable Differential Equations
Equations that can be rearranged so all y-terms are on one side and all x-terms on the other, then integrated separately.
Exponential Growth/Decay Model
The equation dy/dt = k*y has the solution y = y0*e^(kt), where y0 is the initial value and k is the growth (or decay, if negative) rate.
Initial Conditions
A specific starting value (like y(0)=5) needed to pin down one exact solution from the general family of solutions to a differential equation.
02 Key Formulas
- dy/dt = k*y has solution y = y0 * e^(kt)
03 Solved Examples
Example 1 Solve dy/dt = 3y with initial condition y(0)=10.
- This matches the exponential growth form dy/dt=k*y with k=3.
- Apply the solution formula: y = y0*e^(kt) = 10*e^(3t).
Answer: y = 10*e^(3t)
Example 2 A radioactive sample decays according to dy/dt = -0.5*y, with y(0)=100. Find y at t=2.
- Apply the formula: y = 100*e^(-0.5t).
- At t=2: y = 100*e^(-1).
Answer: y is approximately 36.8
Example 3 Classify the order of the differential equation d^2y/dx^2 + 3*dy/dx = 0.
- Identify the highest derivative present.
- d^2y/dx^2 is a second derivative.
Answer: Second-order
04 Practice Questions
1Solve dy/dt=2y with y(0)=5.
y=5*e^(2t)
2What is the order of dy/dx + y = 0?
First-order
3What is the order of d^2y/dx^2 = -y?
Second-order
4What does 'k' represent in dy/dt=k*y?
The growth (or decay) rate
5Why is an initial condition needed to fully solve a differential equation?
The general solution represents a whole family of curves; the initial condition picks out one specific curve
📄 Differential Equations — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.