Basic concepts · Definition

Initial value problem

differential equationsinitial conditions

Before you start you need to know

The general solution leaves free constants; the initial data of a real phenomenon fix them and select a single curve.

Definition

An initial value problem asks for the solution of a differential equation that, in addition, takes given values at a given point.

\[y'=f(x,y),\qquad y(x_0)=y_0\]

We almost always know the state of the system at one instant: the population when the experiment begins, the position and velocity of a mass when released. That information is the initial condition. The procedure is always the same: find the general solution, substitute the point \((x_0,y_0)\) and solve for the constants.

For example, the general solution of \(y'=2y\) is \(y=Ce^{2x}\). With \(y(0)=3\) we get \(C=3\) and the required solution is \(y=3e^{2x}\). Geometrically, the general solution is a family of curves and the initial condition picks the one through the given point.

The number of conditions must match the order: a first-order equation needs one datum \(y(x_0)=y_0\); a second-order one needs two, usually \(y(x_0)\) and \(y'(x_0)\). With fewer conditions some constants stay free, as in \(y''=6x\) with only \(y(0)=1\), which has infinitely many solutions.

Moreover, the solution of an initial value problem lives on an interval containing \(x_0\), and that interval can be small even when the equation is simple: \(y'=y^2\), \(y(0)=1\) gives \(y=1/(1-x)\), valid only on \((-\infty,1)\). It remains to decide when a solution exists and when it is unique, which the next concept answers.

Think about it

How many constants must be fixed in a third-order equation, and which initial data are used?

Three: usually \(y(x_0)\), \(y'(x_0)\) and \(y''(x_0)\).

Exercises 10

  1. Solve \(y'=2y,\ y(0)=3\).
  2. Solve \(y'=2x,\ y(0)=5\).
  3. The general solution of \(y'=-y\) is \(y=Ce^{-x}\). If \(y(0)=4\), what is \(C\)?
  4. Solve \(y'=\cos x,\ y(0)=1\).
  5. Which function solves \(y'=y,\ y(1)=e\)?
  6. If \(y'=3x^2\) and \(y(1)=2\), what is \(y(2)\)?
  7. If \(y'=-2y\) and \(y(0)=1\), what is \(y(\ln 2)\)?
  8. Solve \(y''=6x\) with \(y(0)=1\) and \(y'(0)=2\).
  9. The solution of \(y'=y^2,\ y(0)=1\) is \(y=\frac{1}{1-x}\). What is the largest interval containing \(0\) where it is defined?
  10. How many solutions of \(y''=6x\) satisfy only \(y(0)=1\)?

References

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 12th edition, Wiley, 2021. §1.2
  • Dennis G. Zill, A First Course in Differential Equations with Modeling Applications, 11th edition, Cengage, 2018. §1.2