Key takeaway: A differential equation is an equation that relates an unknown function to one or more of its derivatives, and solving it means finding the functions that make the relation true.
Grade 12 – AP Calculus ABAP Calculus AB Unit 7Calculus
Definition of Differential Equation
A differential equation describes how a quantity changes instead of stating the quantity directly. A statement such as “the rate of change of the amount equals twice the amount” can be written as an equation in a function and its derivative, and the solution is the function itself.
Solutions usually come in families, because an unknown constant survives integration. A starting condition, such as the value at \(x=0\), picks out one member of the family. That single curve is called a particular solution.
Formula and notation
A first-order example, written with the derivative of the unknown \(y\):
\[ \frac{dy}{dx}=-2y+6. \]
The order is the highest derivative that appears. A candidate solution is checked by substituting \(y\) and \(\frac{dy}{dx}\) into both sides and confirming they agree.
Key facts
- The general solution holds an arbitrary constant; the particular solution has a number in its place.
- An initial condition gives one point that the solution curve must pass through.
- Equilibrium solutions are constant functions where the derivative is zero everywhere.
- You can always verify a proposed solution of a differential equation by differentiating and substituting.
Differential Equation example
Show that \(y=4e^{-2x}+3\) solves \(\frac{dy}{dx}=-2y+6\), and find the constant if \(y(0)=7\) is required.
-
Left side: \(\frac{dy}{dx}=4\cdot(-2)e^{-2x}=-8e^{-2x}\).
-
Right side: \(-2(4e^{-2x}+3)+6=-8e^{-2x}-6+6=-8e^{-2x}\). The two sides agree.
-
For the general family \(y=Ce^{-2x}+3\), the condition gives \(C+3=7\), so \(C=4\).
The function satisfies the equation, and the starting value 7 selects \(C=4\).
Common mistake
Watch out
A student claims \(y=4e^{-2x}\) also solves the equation, because its derivative is \(-8e^{-2x}\), and drops the \(+3\).
Why it is wrong
Substituting \(y=4e^{-2x}\) into the right side gives \(-8e^{-2x}+6\), which differs from \(-8e^{-2x}\) by 6. The constant 3 is needed to cancel that 6. A solution of a differential equation must satisfy both sides exactly, not just similar terms.
Related terms and lessons
Browse the glossary: A–Z list · AP Calculus AB vocabulary · Calculus