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What Is a Differential Equation?
What Is a Differential Equation?
A differential equation relates an unknown function to one or more of its derivatives.
Examples
$$y'=3x^2$$, $$\frac{dy}{dx}=ky$$, and $$y''+4y=0$$ are all differential equations.
Order
The order is the highest derivative present. If $y''$ appears but no higher derivative does, the equation is second order.
Initial Condition
An initial condition such as $y(0)=5$ selects one solution from a family.
Example 1
Find the order of $y''+y'=0$.
Second order.
Example 2
Why is $y'=ky$ a growth model when $k>0$?
The rate of change is proportional to the current amount, so positive values tend to increase.
Practice Problems
1. What is the order of $y'''-2y=0$?
2. What does $y(1)=4$ represent?
3. Is $y'=2y$ first order or second order?
4. What does a slope field visualize?
5. Name a real-world process modeled by a differential equation.
Show Answer Key
1. Third order
2. An initial condition or specific function value
3. First order
4. The slope of solution curves at many points
5. Population growth, cooling, circuits, motion, and many others