Calculating Constant C in Abel's Formula for Wronskian

[Clausius2]

When dealing with Abel's formula for the wronskian of a second order ODE:

$$W(R)=Ce^{-\int p_1(R)dR}$$

and assuming that you don't know the homogeneous solutions but you know their asymptotic behavior at infinity and at the origin, how is the constant C calculated?

Thanks.

 

[gammamcc]

Do you know the behavior of the 1st-order derivatives at the origin? You can take C = W(0) it the integral of p_1 has 0 for lower limit. Or, is the problem more involved than that?

 

[tacman]

The constant C in Abel's formula for the wronskian can be calculated by using the known asymptotic behavior of the homogeneous solutions at infinity and at the origin. This can be done by substituting the asymptotic forms of the solutions into the formula and solving for C.

For example, if the asymptotic behavior of the solutions is given by W(R) = A(R) + B(R), with A(R) approaching a constant at infinity and B(R) approaching zero at the origin, then we can write:

W(R) = Ce^{-\int p_1(R)dR} = Ae^{-\int p_1(R)dR} + Be^{-\int p_1(R)dR}

At infinity, the exponential term will dominate and we can neglect the constant term. This gives us:

W(R) = Ae^{-\int p_1(R)dR}

Similarly, at the origin, the constant term will dominate and we can neglect the exponential term. This gives us:

W(R) = B

Equating these two expressions for W(R), we can solve for C:

Ae^{-\int p_1(R)dR} = B

C = \frac{B}{e^{-\int p_1(R)dR}}

Therefore, the constant C can be calculated by using the known asymptotic behavior of the solutions at infinity and at the origin. This method can be applied to any given ODE and its solutions to determine the value of C in Abel's formula for the wronskian.