Differential Equation
In short: Differential equation ties an unknown function to its own rate of change, and solving it means discovering which functions follow that rule at every input.
AP Calculus ABCalculusAP Calculus AB Unit 7
What is Differential Equation?
A differential equation describes how something changes instead of stating its value directly. “The cooler the soup, the slower it loses heat” is a rule about rates, and writing that rule with a derivative produces a differential equation.
A solution is any function that turns the rule into a true statement once you substitute it. Usually a whole family works, the general solution, carrying an arbitrary constant. A starting value such as \(y(0) = 5\) picks out one member, the particular solution. Example: \(\frac{dy}{dx} = 6x^2\) has general solution \(y = 2x^3 + C\).
How to spot it
- The statement contains dy/dx, y’, or dP/dt next to the unknown function itself, as in a differential equation.
- A story says the rate of change is proportional to the amount present, or equal to some expression.
- The task is to confirm that a formula fits the rule by plugging it in.
- An initial condition like y(0) = 40 is given to pick out a single curve.
The method
- To check a proposed solution, compute the derivatives it needs.
- Substitute the function and its derivatives into both sides of the equation.
- If both sides simplify to the same expression for all \(x\), the function is a solution.
- To use an initial condition, put the given point into the general solution and solve for \(C\).
Differential Equation: solved example
Problem
Show that \(y = 5e^{-2x}\) is a solution of \(\frac{dy}{dx} = -2y\), and find the value of \(\frac{dy}{dx}\) when \(x = 0\).
Solution
Differentiate: \(\frac{dy}{dx} = 5 \cdot (-2)e^{-2x} = -10e^{-2x}\).
Now the right side: \(-2y = -2 \cdot 5e^{-2x} = -10e^{-2x}\). The two sides match for every \(x\), so the formula works.
At \(x = 0\), \(y = 5\) and the slope is \(-2(5) = \) \(-10\).
Try it
Question 1
Find the particular solution of \(\frac{dy}{dx} = 6x^2\) with \(y(0) = 7\).
Show the answer
Integrate: \(y = 2x^3 + C\). Then \(y(0) = C = 7\), so \(y = 2x^3 + 7\).
Question 2
Is \(y = \sin x\) a solution of \(y^{\prime\prime} + y = 0\)?
Show the answer
Yes. \(y^{\prime\prime} = -\sin x\), so \(y^{\prime\prime} + y = -\sin x + \sin x = 0\).
Differential Equation vs. Slope Field
The equation is the rule written in symbols. A slope field is a picture of that rule, with a little slope mark drawn at many points. Solution curves follow the marks, but the field alone does not give a formula for the function.
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