solving an implicit ordinary differential equation

I have to differential equations and I'm trying to solve these using ode45 or ode15 in Matlab:

dP/dt= f(R,P,dR/dt)

dR/dt=f(R,P)

any idea how I can solve these two equations ?

Answer 1

[David Goodmanson]

Hi Nariman,

I assume that the two functions you show as f are not the same function, so I will denote the second one as g. And you don't say whether P and R are scalars, so I assume they are column vectors of size m and n respectively. The input to ode45 is a column vector [P ; R] of length m+n, and so is the output vector [dP/dt ; dR/dt]. You have

dP/dt = f(R,P,dR/dt)
dR/dt = g(P,R)

which, using substitution and composition of functions is the same as

dP/dt = f(R,P,g(P,R))
dR/dt = g(P,R)

so the output d/dt vector is determined by these equations.

[Nariman Mahabadi]

Hi David,

Thank you for the answer. The equations are:

dP/dt=6R/(PR^3) - 3P/R*dR/dt

dR/dt=R/4*(P-2/R)

the Initial conditions: P=10; and R=20;

The time period to solve these equations is [0 100]

So is this the correct form?

f = @(t,x) [6*x(2)/(x(1)*x(2)^3) - 3*x(1)/x(2)*dR/dt ; x(2)/4*(x(1)-2/x(2))]

[t,xa] = ode45(f,[0 100],[10 20]);

[David Goodmanson]

Hi Nariman, you should run this and see. It's a valuable thing to do because the error message is very informative, and otherwise you are operating in a vacuum.

When you run it, you will find that f has no way of knowing what dR/dT is, since it is not supplied on input. (Since it is in dR/dT form and not denoted as a single variable such as dRdT, f does not know what either of dR or dT are. So it coughs up and error message about dR). In place of dR/dT you need to insert the equivalent expression in terms of x(1) and x(2), but you already have that.