where P, Q are functions of x only; this is known as the linear equation, since it contains y and dy/dx only to the first degree. If ƒPdx = u, we clearly have
| d | (yeu) = eu | ( | dy | + Py | ) | = euQ, |
| dx | dx |
so that y = e-u(ƒeuQdx + A) solves the equation, and is the only possible solution, A being an arbitrary constant. The rule for the solution of the linear equation is thus to multiply the equation by eu, where u = ƒPdx.
A third simple and important form is that denoted by
y = px + ƒ(p),
where p is an abbreviation for dy/dx; this is known as Clairaut’s form. By differentiation in regard to x it gives
| p = p + x | dp | + ƒ′(p) | dp | , |
| dx | dx |
where
| ƒ′(p) = | d | ƒ(p); |
| dp |
thus, either (i.) dp/dx = 0, that is, p is constant on the curve satisfying the differential equation, which curve is thus any one of the straight lines y = cx = ƒ(c), where c is an arbitrary constant, or else, (ii.) x + ƒ′(p) = 0; if this latter hypothesis be taken, and p be eliminated between x + ƒ′(p) = 0 and y = px + ƒ(p), a relation connecting x and y, not containing an arbitrary constant, will be found, which obviously represents the envelope of the straight lines y = cx + ƒ(c).