AP® Calculus BC · Unit 7 · Topic 7.2
Verifying Solutions for Differential Equations
Verifying a solution means plugging a candidate function and its derivative into the differential equation and checking both sides come out equal for every x, not just one number.
A differential equation describes a relationship between a function and its own rate of change, so a solution isn't a number, it's an entire function. To check that y = f(x) solves dy/dx = g(x, y), you find dy/dx from f(x), then substitute both f(x) and that derivative into the equation. If the left side matches the right side for every x, the function checks out. This is the same idea as checking x = 5 solves 2x + 1 = 11, just one level up: instead of plugging a number into an algebraic equation, you plug a function and its derivative into a differential equation.
The substitution has to happen on both sides at once. Find dy/dx first, then replace every y in the original equation with your candidate function, and replace dy/dx with the derivative you just computed. Simplify each side independently and see if they agree. Watch what happens with a constant: differentiating y = x2 + C gives dy/dx = 2x regardless of C, since the derivative of any constant is zero. That single fact explains why the equation dy/dx = 2x is satisfied by infinitely many functions, one for every value of C, forming a family of solutions that all share the same slope behavior.
This sets up the next idea in the unit: slope fields. A slope field is a picture of that entire family of solutions at once, drawn without solving anything, just by reading the slope g(x, y) at a grid of points. Verifying a solution tells you a particular curve fits the pattern; a slope field shows you the whole pattern. Later, an initial condition like y(0) = 7 picks out exactly one curve from the family, turning the general solution into a particular one.
Key terms
- differential equation
- An equation relating a function to one or more of its derivatives.
- particular solution
- One specific function from the solution family, pinned down by an initial condition.
- general solution
- The entire family of functions, usually written with an arbitrary constant C, that satisfies the differential equation.
- initial condition
- A given point, such as y(0) = 7, used to select one particular solution from the general solution.
Watch it explained
- How it worksHow do I substitute a candidate function and its derivative into both sides at once?
- Worked exampleShow me that y = 3e2x really does solve dy/dx = 2y.
- Worked exampleShow me why y = x2 + C solves dy/dx = 2x no matter what C is.
- Common mistakeWhy doesn't a differential equation have exactly one solution?
Check yourself
End of topic.