Unable to find explicit solution to differential eq

Hi there!

Pardon me, I'm new to differential equations and particularly in MATLAB, and I have stumbled upon an error that I don't understand the reasoning behind.

I'm getting the error "Warning: Unable to find explicit solution."

kp = 2*10^7;
km = 0.27*10^-4;

syms x(t) x2(t);

ode1 = diff(x,t) == 2*km*x2(t) - 2*kp*x(t)^2;
ode2 = diff(x2,t) == -km*x2(t) + kp*x(t)^2;
odes = [ode1, ode2];

cond1 = x(0) == 10^-9;
cond2 = x2(0) == 0;
conds = [cond1, cond2];

S = dsolve(odes,conds)

It works fine whenever i remove the squaring of x(t) in both differential equations.

Is this caused by some inner workings of the dsolve function?

[John D'Errico]

And you know that an analytical solution exists? When you remove the squares, the system is now linear, so a solution is trivial.

[Torsten]

Note that

diff(x,t)+2*diff(x2,t)=0,

thus

x(t)+2*x2(t)=x(0)+2*x2(0)=1e-9

and so

x2(t)=5e-10-0.5*x(t).

Insert this expression in one of the differential equation and solve by separation of variables.

[Ditlev Bornebusch]

Thank you for the tip.

To solve by separation of variables the function has to be on the form:

g(y)*dy/dx=f(x)

Where the function g(y) is independent of x and f (x) is independent of y.

The variable is t for both function x2 and x, so I don't see how you would perform separation of variables.

I had reduced a system of differential equations to the initial problem for the sake of simplicity and here is the full system (the constants k's, are set to an arbitrary value for this example)

k1 = 1;      
k2 = 2;           
k3 = 3;      
k4 = 4; 
k5 = 5;      
k6 = 6; 
k7 = 7;      
k8 = 8;   
k9 = 9; 

syms x1(t) x2(t) x3(t) x4(t) x5(t);

ode1 = diff(x1,t) == -(k3*k1 + k5*k2)*x1(t) + k4*x2(t) + k6*x3(t) + 2*k8*x5(t) - 2*k7*x1(t)^2;
ode2 = diff(x2,t) == k3*k1*x1(t) - (k4 + k5*k2)*x2(t) + k6*x4(t);
ode3 = diff(x3,t) == k5*k2*x1(t) - (k6 + k3*k1)*x3(t) + k4*x4(t);
ode4 = diff(x4,t) == k5*k2*x2(t) + k3*k1*x3(t) - (k4 + k6 + k9)*x4(t);
ode5 = diff(x5,t) == k9*x4(t) - k8*x5(t) + k7*x1(t)^2;

cond1 = x1(0) == 10;
cond2 = x2(0) == 0;
cond3 = x3(0) == 0;
cond4 = x4(0) == 0;
cond5 = x5(0) == 0;
conds = [cond1; cond2; cond3; cond4; cond5];

S = dsolve(odes, conds)

Although I am also wondering, why MATLAB cannot handle that non-linearity in finding a solution.

Mathematica for instance has no issues with this.

Is this a property of the dsolve function of MATLAB and is there another method in MATLAB, that can solve a large system of non-linear differential equations?

[Matt J]

Mathematica for instance has no issues with this.

I'm willing to bet that you haven't entered the same ODE in both Matlab and Mathematica. As a test, you could take the solution returned by Mathematica and plug it into the ODE as implemented in Matlab to see if the equations (as implemented in Matlab) are satisfied.

[Walter Roberson]

Remember that Mathematics converts floating point constants a different way that MATLAB does.

[Walter Roberson]

If you switch the numeric values into symbolic constants then it turns out there is a series of four analytic solutions.... that are quite long. Roots of a quartic are involved, multiple times. One of the shorter ways to write the expressions involve tanh(); it is also possible to rewrite the tanh() in terms of log() of a complex (and complicated) expression. Or you can rewrite in terms of exponentials, but that gets rather long. I gave up trying to simplfy the expressions.

I suspect that if you were processing the same expressions that Mathematica would probably be willing to produce an answer, but that the answer would not be simple at all.

These kinds of systems are beyond the abilities of MATLAB.

Answer 1

[David Goodmanson]

Hi Ditlev,

I believe this does have an analytic solution that is not overly complicated, of the form

x = (a+be^(-ct)) / (f+ge^(-ct))

First of all the scaling of this problem is not too good so I went to the variables (I used y in place of your x2)

u = km*t         x = (kp/km) x_old        y = (kp/km) y_old

in which case the equations are

x' = 2y - 2x^2
y' = -y + x^2

where differentiation is d/du. The starting value for the new x is

(kp/km)*1e-9 = 740.7

The most complicated the algebra gets is the roots of a quadratic eqn. The code below plots both a numerical and analytic solution.

Eventually the solution gets to the fixed point

x = r1 = 18.9966      y = -r1/2 + C = 360.8721

at which point x^2 = y and both derivatives are zero.

kp = 2*10^7;
km = 0.27*10^-4;
x0 = (kp/km)*1e-9   % starting point for x
y0 = 0;

% numerical version
[unum xynum] = ode45(@(u,xy) fun(u,xy), [0 .01], [x0 y0]);         % xy = [x; y]

%analytic version
C = x0/2;
r = roots([1,1/2,-C]);
r1 = max(r);
r2 = min(r);
alpha = 2*(r1-r2);
D = (x0-r1)/(x0-r2);
u = 0:1e-4:.01;
x = (r1-D*r2*exp(-alpha*u))./(1-D*exp(-alpha*u));   % solution
y = -x/2 + C;

figure(1)
plot(unum,xynum,u,x,'o',u,y,'o')
grid on

function dxy = fun(u,xy)
  dxy = zeros(2,1);
  dxy(1) = 2*xy(2)-2*xy(1)^2;
  dxy(2) = -xy(2) + xy(1)^2;
end

.

[Walter Roberson]

One of the solutions involves

x(t) = 1/4*(tanh(2*(km*(8*ic1*kp+km))^(1/2)*(1/8*(ic1*kp*t+1/4*km*t+1/2*ln(1/4/ic1/kp*2^(1/2)*(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4)^(1/2))^(1/2)))/ic1^2/kp^2*(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4)^(1/2))+ic1*kp*t+1/4*km*t-1/2*ln(1/4/ic1/kp*2^(1/2)*(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4)^(1/2))^(1/2)))/(1/8/ic1^2/kp^2*(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4)^(1/2))+1)/(4*ic1*kp+km))*(km*(8*ic1*kp+km))^(1/2)-km)/kp

which can be rewritten without the tanh() as

x(t) = -1/2*km*(1/2*2^(1/2)*(-(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+km*(8*ic1*kp+km))*(exp((-I*2^(1/2)*(-(4*ic1*kp+km)*((4*ic1*kp+km)*((-8*ic1^2*kp^2-8*ic1*km*kp-km^2)*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4)*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))^2+((-4*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(1/ic1/kp)+6*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(2)-2*t*(32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+2*(4*ic1*kp+km)*((64*ic1^4*kp^4+256*ic1^3*km*kp^3+160*ic1^2*km^2*kp^2+32*ic1*km^3*kp+2*km^4)*ln(1/ic1/kp)+(-96*ic1^4*kp^4-384*ic1^3*km*kp^3-240*ic1^2*km^2*kp^2-48*ic1*km^3*kp-3*km^4)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)))*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))-((16*ic1^2*kp^2+16*ic1*km*kp+2*km^2)*ln(1/ic1/kp)+(-24*ic1^2*kp^2-24*ic1*km*kp-3*km^2)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km))*((8*ic1*kp+2*km)*ln(1/ic1/kp)+(-12*ic1*kp-3*km)*ln(2)+t*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+((128*ic1^4*kp^4+512*ic1^3*km*kp^3+320*ic1^2*km^2*kp^2+64*ic1*km^3*kp+4*km^4)*ln(1/ic1/kp)^2+((-384*ic1^4*kp^4-1536*ic1^3*km*kp^3-960*ic1^2*km^2*kp^2-192*ic1*km^3*kp-12*km^4)*ln(2)+4*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*ln(1/ic1/kp)+(288*ic1^4*kp^4+1152*ic1^3*km*kp^3+720*ic1^2*km^2*kp^2+144*ic1*km^3*kp+9*km^4)*ln(2)^2-6*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(2)+t^2*km*(8*ic1*kp+km)*(32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4))*(4*ic1*kp+km)))^(1/2)+I*(4*ic1*kp+km)*Pi*(8*ic1^2*kp^2+8*ic1*km*kp+km^2-(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)))/(4*ic1*kp+km)/(8*ic1^2*kp^2+8*ic1*km*kp+km^2-(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)))-1)*((4*ic1*kp+km)*((4*ic1*kp+km)*((-8*ic1^2*kp^2-8*ic1*km*kp-km^2)*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4)*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))^2+((-4*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(1/ic1/kp)+6*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(2)-2*t*(32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+2*(4*ic1*kp+km)*((64*ic1^4*kp^4+256*ic1^3*km*kp^3+160*ic1^2*km^2*kp^2+32*ic1*km^3*kp+2*km^4)*ln(1/ic1/kp)+(-96*ic1^4*kp^4-384*ic1^3*km*kp^3-240*ic1^2*km^2*kp^2-48*ic1*km^3*kp-3*km^4)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)))*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))-((16*ic1^2*kp^2+16*ic1*km*kp+2*km^2)*ln(1/ic1/kp)+(-24*ic1^2*kp^2-24*ic1*km*kp-3*km^2)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km))*((8*ic1*kp+2*km)*ln(1/ic1/kp)+(-12*ic1*kp-3*km)*ln(2)+t*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+((128*ic1^4*kp^4+512*ic1^3*km*kp^3+320*ic1^2*km^2*kp^2+64*ic1*km^3*kp+4*km^4)*ln(1/ic1/kp)^2+((-384*ic1^4*kp^4-1536*ic1^3*km*kp^3-960*ic1^2*km^2*kp^2-192*ic1*km^3*kp-12*km^4)*ln(2)+4*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*ln(1/ic1/kp)+(288*ic1^4*kp^4+1152*ic1^3*km*kp^3+720*ic1^2*km^2*kp^2+144*ic1*km^3*kp+9*km^4)*ln(2)^2-6*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(2)+t^2*km*(8*ic1*kp+km)*(32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4))*(4*ic1*kp+km)))^(1/2)+I*(8*ic1*kp+km)*(((-24*ic1^2*kp^2-16*ic1*km*kp-2*km^2)*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+2*(4*ic1^2*kp^2+8*ic1*km*kp+km^2)*(4*ic1*kp+km)^2)*ln((-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))^(1/2)/ic1/kp)+((36*ic1^2*kp^2+24*ic1*km*kp+3*km^2)*ln(2)-t*(4*ic1*kp+km)*(4*ic1^2*kp^2+8*ic1*km*kp+km^2))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+(4*ic1*kp+km)*((-48*ic1^3*kp^3-108*ic1^2*km*kp^2-36*ic1*km^2*kp-3*km^3)*ln(2)+t*km*(6*ic1*kp+km)*(2*ic1*kp+km)*(8*ic1*kp+km)))*(exp(1/2*(-2^(1/2)*((4*ic1*kp+km)*((4*ic1*kp+km)*((-8*ic1^2*kp^2-8*ic1*km*kp-km^2)*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4)*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))^2+((-4*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(1/ic1/kp)+6*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(2)-2*t*(32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+2*(4*ic1*kp+km)*((64*ic1^4*kp^4+256*ic1^3*km*kp^3+160*ic1^2*km^2*kp^2+32*ic1*km^3*kp+2*km^4)*ln(1/ic1/kp)+(-96*ic1^4*kp^4-384*ic1^3*km*kp^3-240*ic1^2*km^2*kp^2-48*ic1*km^3*kp-3*km^4)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)))*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))-((16*ic1^2*kp^2+16*ic1*km*kp+2*km^2)*ln(1/ic1/kp)+(-24*ic1^2*kp^2-24*ic1*km*kp-3*km^2)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km))*((8*ic1*kp+2*km)*ln(1/ic1/kp)+(-12*ic1*kp-3*km)*ln(2)+t*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+((128*ic1^4*kp^4+512*ic1^3*km*kp^3+320*ic1^2*km^2*kp^2+64*ic1*km^3*kp+4*km^4)*ln(1/ic1/kp)^2+((-384*ic1^4*kp^4-1536*ic1^3*km*kp^3-960*ic1^2*km^2*kp^2-192*ic1*km^3*kp-12*km^4)*ln(2)+4*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*ln(1/ic1/kp)+(288*ic1^4*kp^4+1152*ic1^3*km*kp^3+720*ic1^2*km^2*kp^2+144*ic1*km^3*kp+9*km^4)*ln(2)^2-6*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(2)+t^2*km*(8*ic1*kp+km)*(32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4))*(4*ic1*kp+km)))^(1/2)-I*2^(1/2)*(-(4*ic1*kp+km)*((4*ic1*kp+km)*((-8*ic1^2*kp^2-8*ic1*km*kp-km^2)*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4)*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))^2+((-4*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(1/ic1/kp)+6*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(2)-2*t*(32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+2*(4*ic1*kp+km)*((64*ic1^4*kp^4+256*ic1^3*km*kp^3+160*ic1^2*km^2*kp^2+32*ic1*km^3*kp+2*km^4)*ln(1/ic1/kp)+(-96*ic1^4*kp^4-384*ic1^3*km*kp^3-240*ic1^2*km^2*kp^2-48*ic1*km^3*kp-3*km^4)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)))*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))-((16*ic1^2*kp^2+16*ic1*km*kp+2*km^2)*ln(1/ic1/kp)+(-24*ic1^2*kp^2-24*ic1*km*kp-3*km^2)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km))*((8*ic1*kp+2*km)*ln(1/ic1/kp)+(-12*ic1*kp-3*km)*ln(2)+t*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+((128*ic1^4*kp^4+512*ic1^3*km*kp^3+320*ic1^2*km^2*kp^2+64*ic1*km^3*kp+4*km^4)*ln(1/ic1/kp)^2+((-384*ic1^4*kp^4-1536*ic1^3*km*kp^3-960*ic1^2*km^2*kp^2-192*ic1*km^3*kp-12*km^4)*ln(2)+4*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*ln(1/ic1/kp)+(288*ic1^4*kp^4+1152*ic1^3*km*kp^3+720*ic1^2*km^2*kp^2+144*ic1*km^3*kp+9*km^4)*ln(2)^2-6*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(2)+t^2*km*(8*ic1*kp+km)*(32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4))*(4*ic1*kp+km)))^(1/2)+I*(4*ic1*kp+km)*Pi*(8*ic1^2*kp^2+8*ic1*km*kp+km^2-(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)))/(4*ic1*kp+km)/(8*ic1^2*kp^2+8*ic1*km*kp+km^2-(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)))-exp(1/2*(2^(1/2)*((4*ic1*kp+km)*((4*ic1*kp+km)*((-8*ic1^2*kp^2-8*ic1*km*kp-km^2)*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4)*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))^2+((-4*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(1/ic1/kp)+6*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(2)-2*t*(32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+2*(4*ic1*kp+km)*((64*ic1^4*kp^4+256*ic1^3*km*kp^3+160*ic1^2*km^2*kp^2+32*ic1*km^3*kp+2*km^4)*ln(1/ic1/kp)+(-96*ic1^4*kp^4-384*ic1^3*km*kp^3-240*ic1^2*km^2*kp^2-48*ic1*km^3*kp-3*km^4)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)))*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))-((16*ic1^2*kp^2+16*ic1*km*kp+2*km^2)*ln(1/ic1/kp)+(-24*ic1^2*kp^2-24*ic1*km*kp-3*km^2)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km))*((8*ic1*kp+2*km)*ln(1/ic1/kp)+(-12*ic1*kp-3*km)*ln(2)+t*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+((128*ic1^4*kp^4+512*ic1^3*km*kp^3+320*ic1^2*km^2*kp^2+64*ic1*km^3*kp+4*km^4)*ln(1/ic1/kp)^2+((-384*ic1^4*kp^4-1536*ic1^3*km*kp^3-960*ic1^2*km^2*kp^2-192*ic1*km^3*kp-12*km^4)*ln(2)+4*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*ln(1/ic1/kp)+(288*ic1^4*kp^4+1152*ic1^3*km*kp^3+720*ic1^2*km^2*kp^2+144*ic1*km^3*kp+9*km^4)*ln(2)^2-6*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(2)+t^2*km*(8*ic1*kp+km)*(32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4))*(4*ic1*kp+km)))^(1/2)-I*2^(1/2)*(-(4*ic1*kp+km)*((4*ic1*kp+km)*((-8*ic1^2*kp^2-8*ic1*km*kp-km^2)*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4)*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))^2+((-4*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(1/ic1/kp)+6*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(2)-2*t*(32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+2*(4*ic1*kp+km)*((64*ic1^4*kp^4+256*ic1^3*km*kp^3+160*ic1^2*km^2*kp^2+32*ic1*km^3*kp+2*km^4)*ln(1/ic1/kp)+(-96*ic1^4*kp^4-384*ic1^3*km*kp^3-240*ic1^2*km^2*kp^2-48*ic1*km^3*kp-3*km^4)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)))*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))-((16*ic1^2*kp^2+16*ic1*km*kp+2*km^2)*ln(1/ic1/kp)+(-24*ic1^2*kp^2-24*ic1*km*kp-3*km^2)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km))*((8*ic1*kp+2*km)*ln(1/ic1/kp)+(-12*ic1*kp-3*km)*ln(2)+t*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+((128*ic1^4*kp^4+512*ic1^3*km*kp^3+320*ic1^2*km^2*kp^2+64*ic1*km^3*kp+4*km^4)*ln(1/ic1/kp)^2+((-384*ic1^4*kp^4-1536*ic1^3*km*kp^3-960*ic1^2*km^2*kp^2-192*ic1*km^3*kp-12*km^4)*ln(2)+4*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*ln(1/ic1/kp)+(288*ic1^4*kp^4+1152*ic1^3*km*kp^3+720*ic1^2*km^2*kp^2+144*ic1*km^3*kp+9*km^4)*ln(2)^2-6*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2)*ln(2)+t^2*km*(8*ic1*kp+km)*(32*ic1^4*kp^4+128*ic1^3*km*kp^3+80*ic1^2*km^2*kp^2+16*ic1*km^3*kp+km^4))*(4*ic1*kp+km)))^(1/2)+I*(4*ic1*kp+km)*Pi*(8*ic1^2*kp^2+8*ic1*km*kp+km^2-(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)))/(4*ic1*kp+km)/(8*ic1^2*kp^2+8*ic1*km*kp+km^2-(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)))))/((4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2-(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))*((4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2-(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))^2+(((-16*ic1*kp-4*km)*ln(1/ic1/kp)+(24*ic1*kp+6*km)*ln(2)-2*t*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+2*(4*ic1*kp+km)*((16*ic1^2*kp^2+16*ic1*km*kp+2*km^2)*ln(1/ic1/kp)+(-24*ic1^2*kp^2-24*ic1*km*kp-3*km^2)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km)))*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))-(4*ic1*kp*t+km*t-3*ln(2)+2*ln(1/ic1/kp))*((8*ic1*kp+2*km)*ln(1/ic1/kp)+(-12*ic1*kp-3*km)*ln(2)+t*km*(8*ic1*kp+km))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+(4*ic1*kp+km)*((32*ic1^2*kp^2+32*ic1*km*kp+4*km^2)*ln(1/ic1/kp)^2+((-96*ic1^2*kp^2-96*ic1*km*kp-12*km^2)*ln(2)+4*t*km*(8*ic1*kp+km)*(4*ic1*kp+km))*ln(1/ic1/kp)+(72*ic1^2*kp^2+72*ic1*km*kp+9*km^2)*ln(2)^2-6*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*ln(2)+t^2*km*(8*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))))^(1/2)/(8*ic1*km*kp+km^2-(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))/(exp(I*(-32*Pi*ic1^3*kp^3-40*Pi*ic1^2*km*kp^2-12*Pi*ic1*km^2*kp+4*Pi*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)*ic1*kp-Pi*km^3+Pi*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)*km+(-(4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2-(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))*((4*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2-(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))^2+(((-16*ic1*kp-4*km)*ln(1/ic1/kp)+(24*ic1*kp+6*km)*ln(2)-2*t*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+2*(4*ic1*kp+km)*((16*ic1^2*kp^2+16*ic1*km*kp+2*km^2)*ln(1/ic1/kp)+(-24*ic1^2*kp^2-24*ic1*km*kp-3*km^2)*ln(2)+t*km*(8*ic1*kp+km)*(4*ic1*kp+km)))*ln(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2))-(4*ic1*kp*t+km*t-3*ln(2)+2*ln(1/ic1/kp))*((8*ic1*kp+2*km)*ln(1/ic1/kp)+(-12*ic1*kp-3*km)*ln(2)+t*km*(8*ic1*kp+km))*(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)+(4*ic1*kp+km)*((32*ic1^2*kp^2+32*ic1*km*kp+4*km^2)*ln(1/ic1/kp)^2+((-96*ic1^2*kp^2-96*ic1*km*kp-12*km^2)*ln(2)+4*t*km*(8*ic1*kp+km)*(4*ic1*kp+km))*ln(1/ic1/kp)+(72*ic1^2*kp^2+72*ic1*km*kp+9*km^2)*ln(2)^2-6*t*km*(8*ic1*kp+km)*(4*ic1*kp+km)*ln(2)+t^2*km*(8*ic1*kp+km)*(8*ic1^2*kp^2+8*ic1*km*kp+km^2))))^(1/2))/(4*ic1*kp+km)/(-8*ic1^2*kp^2-8*ic1*km*kp-km^2+(km*(8*ic1*kp+km)*(4*ic1*kp+km)^2)^(1/2)))-1)/kp

In the above, ic1 is the initial condition cond1

[Ditlev Bornebusch]

This suffices for my current work.

Thank you all for your help.