derivative as parameter for solving differential equations

The solution of some differential equationsMathworldPlanetmath of the forms  x=f(dydx)  and  y=f(dydx)  may be expressed in a parametric form by taking for the parameter the derivative

p:=dydx. (1)

I.  Consider first the equation

x=f(dydx), (2)

for which we suppose that  pf(p)  and its derivative  pf(p)  are continuousMathworldPlanetmath and  f(p)0  on an interval[p1,p2].  It follows that on the interval, the functionMathworldPlanetmathpf(p)  changes monotonically from  f(p1):=x1  to  f(p2):=x2, whence conversely the equation

x=f(p) (3)

defines from  [p1,p2]  onto  [x1,x2]  a bijection

p=g(x) (4)

which is continuously differentiable.  Thus on the interval  [x1,x2],  the differential equation (2) can be replaced by the equation

dydx=g(x), (5)

and therefore, the solution of (2) is

y=g(x)𝑑x+C. (6)

If we cannot express g(x) in a , we take p as an independent variable through the substitution (3), which maps  [x1,x2]  bijectively onto  [p1,p2].  Then (6) becomes a function of p, and by the chain ruleMathworldPlanetmath,


Accordingly, the solution of the given differential equation may be presented on  [p1,p2]  as

{x=f(p),y=pf(p)𝑑p+C. (7)

II.  With corresponding considerations, one can write the solution of the differential equation

y=f(dydx):=f(p), (8)

where p changes on some interval  [p1,p2]  where f(p) and f(p) are continuous and  pf(p)0,  in the parametric presentation

{x=f(p)p𝑑p+C,y=f(p). (9)

III.  The procedures of I and II may be generalised for the differential equations of  x=f(y,p)  and  y=f(x,p);  let’s consider the former one.


x=f(y,p) (10)

we regard y as the independent variable and differentiate with respect to it:


Supposing that the partial derivativeMathworldPlanetmathfp(y,p) does not vanish identically, we get

dpdy=1p-fy(y,p)fp(y,p):=g(y,p). (11)

If  p=p(y,C)  is the general solution of (11), we obtain the general solution of (10):

x=f(y,p(y,C)) (12)
Title derivative as parameter for solving differential equations
Canonical name DerivativeAsParameterForSolvingDifferentialEquations
Date of creation 2013-03-22 18:28:39
Last modified on 2013-03-22 18:28:39
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Topic
Classification msc 34A05
Related topic InverseFunctionTheorem
Related topic ImplicitFunctionTheorem
Related topic DAlembertsEquation
Related topic ClairautsEquation