Solutions and solution families · Differential Equations

Intuition

Integrating once introduces one arbitrary constant, so a first-order equation does not have a solution — it has a whole family of them, one for each value of that constant. Drawn on the plane the family fills up the picture: a stack of curves that never touch, covering every point exactly once. Solving is finding the family; picking a member of it is a second step, and it needs a second piece of information.

Knowing your speed for the whole journey does not tell anyone where you are, because it never says where you started. Every possible starting point gives a different journey, and they are all consistent with the same speedometer reading. The constant of integration is the starting point that the equation never mentions.

The family $y=Ce^{x}$ , which is every solution of $y'=y$ . Positive $C$ above the axis, negative below, and the constant zero between them; no two curves of the family meet.

General, particular, explicit, implicit

The general solution of an equation is the family holding every solution, written with arbitrary constants — one for each order. A particular solution is one member of that family, with the constants fixed to numbers. A solution written as $y=f(x)$ is explicit; one left as a relation $F(x,y)=0$ that cannot be untangled is implicit, and is still a solution.

y'=y\ \Longrightarrow\ y=Ce^{x}

What a family is like

An $n$ th-order equation has $n$ arbitrary constants in its general solution. $y''=0$ gives $y=C_{1}x+C_{2}$ , a two-parameter family.

Applications

Practice

One Constant for Each Order

Each integration introduces one arbitrary constant, so an equation of order n has a general solution with n of them.

y''=0\ \Longrightarrow\ y=C_{1}x+C_{2}

Try it

How many arbitrary constants does the general solution of a third-order equation have?