An initial value problem, or IVP, asks for a function that satisfies a differential equation and also passes through a given starting point. This matters because many real situations, such as motion, cooling, and population growth, are not fully described by a rate rule alone. The same rate equation can produce a whole family of possible functions.
The initial condition selects the one function that matches the actual starting information.
In basic calculus, solving an IVP often means integrating to find a general solution with an unknown constant C. The initial condition, such as y(0) = 5 or s(2) = 10, is then substituted into the general solution to solve for C. Once C is known, the general solution becomes a specific solution.
Graphically, the initial condition is the point that chooses one curve from the family of solution curves.
Key Facts
- An initial value problem includes a differential equation and a condition such as y(a) = b.
- A general solution contains an arbitrary constant, often written as C.
- If dy/dx = f(x), then y = ∫f(x) dx + C.
- Use the initial condition y(a) = b by substituting x = a and y = b into the general solution.
- Example: if dy/dx = 2x, then y = x^2 + C, and y(3) = 10 gives C = 1, so y = x^2 + 1.
- A specific solution is a single function, while a general solution represents a family of functions.
Vocabulary
- Initial value problem
- An initial value problem is a problem that asks for a function satisfying a differential equation and a given starting condition.
- Initial condition
- An initial condition is a known value of the function at a particular input, such as y(0) = 4.
- General solution
- A general solution is a family of functions that satisfies a differential equation and includes one or more arbitrary constants.
- Constant of integration
- The constant of integration is the unknown constant added after taking an indefinite integral.
- Specific solution
- A specific solution is the single function found after using the initial condition to determine the constant.
Common Mistakes to Avoid
- Forgetting the + C after integrating is wrong because the constant represents all vertical shifts of the solution family.
- Substituting the initial condition into the derivative instead of the function is wrong because y(a) = b describes a point on the solution curve, not usually a slope.
- Solving for C before integrating is wrong because C appears in the general solution after the antiderivative is found.
- Leaving the answer as y = F(x) + C is wrong for an IVP because the initial condition should be used to find a numerical value for C.
Practice Questions
- 1 Solve the initial value problem dy/dx = 6x^2 with y(1) = 5.
- 2 A particle has velocity v(t) = 4t - 3 and position s(0) = 10. Find s(t).
- 3 Explain why the differential equation dy/dx = x has many general solutions but the initial value problem dy/dx = x, y(2) = 7 has only one specific solution.