Your d_H_tf_C(tf)==0 turns out to be a differential equation involving x(tf) . You can solve the remaining 4 equations to derive C1, C2, C3, C4, and substitute that in to d_H_tf_C(tf)==0, leaving you with a single differential equation in terms of tf and x(tf) .
When solve() sees a differential equation, it automatically invokes dsolve() on your behalf, but you could make the steps more clear by doing that solve for C1, C2, C3, C4, and trying the dsolve() on what remains.
Unfortunately what remains is too complicated for MATLAB to be able to solve.
When I pass it over to Maple, Maple generates a solution in terms of two indefinite integrals and an unresolved boundary condition. The two indefinite integrals do not appear to have closed form solutions.
EQN = -19440*exp(tf)*(-8/81*exp(3*tf)*(10*x(tf)+diff(x(tf),tf)-30)*cos(3*tf)^4+8/27*( ...
sin(3*tf)*exp(3*tf)+3/2*exp(2*tf)-3/2*exp(4*tf))*diff(x(tf),tf)*cos(3*tf)^3+((( ...
-80/9+40/27*x(tf)+4/27*diff(x(tf),tf))*exp(2*tf)-4/9*(-20+10/3*x(tf)+diff(x(tf) ...
,tf))*exp(4*tf))*sin(3*tf)+(520/81*x(tf)+16/81*diff(x(tf),tf)+200/27)*exp(3*tf) ...
+(-20/9*x(tf)+2/9*diff(x(tf),tf)-20/3)*exp(5*tf)-2/9*exp(tf)*(10*x(tf)+diff(x( ...
tf),tf)+30))*cos(3*tf)^2+2/3*diff(x(tf),tf)*((exp(tf)-22/9*exp(3*tf)+exp(5*tf)) ...
*sin(3*tf)-31/6*exp(2*tf)+31/6*exp(4*tf)-3/2*exp(6*tf)+3/2)*cos(3*tf)+((-49/27* ...
diff(x(tf),tf)-310/27*x(tf)+80/9)*exp(2*tf)+(310/27*x(tf)+25/9*diff(x(tf),tf)- ...
80/9)*exp(4*tf)+(-10/3*x(tf)-diff(x(tf),tf))*exp(6*tf)+10/3*x(tf)+1/3*diff(x(tf ...
),tf))*sin(3*tf)+(-280/27+235/81*diff(x(tf),tf)-440/81*x(tf))*exp(3*tf)+(20/3+ ...
20/9*x(tf)-29/9*diff(x(tf),tf))*exp(5*tf)+exp(7*tf)*diff(x(tf),tf)-7/9*exp(tf)* ...
(-60/7-20/7*x(tf)+diff(x(tf),tf)))/(4*exp(2*tf)*cos(3*tf)^2-22*exp(2*tf)+9*exp( ...
4*tf)+9)^2 == 0
and Maple's dsolve() says
x(tf) = (300*int(-2/27*(cos(3*tf)+1)*(cos(3*tf)-1)*(-1/3*exp(2*tf)*cos(3*tf)^2+ ...
(exp(tf)-exp(3*tf))*sin(3*tf)-7/6*exp(2*tf)+3/4*exp(4*tf)+3/4)*exp((2+3*i)*tf)/ ...
(-4/27*cos(3*tf)^4*exp(3*tf)+(4/9*sin(3*tf)*exp(3*tf)+2/3*exp(2*tf)-2/3*exp(4* ...
tf))*cos(3*tf)^3+((2/9*exp(2*tf)-2/3*exp(4*tf))*sin(3*tf)-1/3*exp(tf)+8/27*exp( ...
3*tf)+1/3*exp(5*tf))*cos(3*tf)^2+((exp(tf)-22/9*exp(3*tf)+exp(5*tf))*sin(3*tf)- ...
31/6*exp(2*tf)+31/6*exp(4*tf)-3/2*exp(6*tf)+3/2)*cos(3*tf)+(-49/18*exp(2*tf)+25 ...
/6*exp(4*tf)-3/2*exp(6*tf)+1/2)*sin(3*tf)-7/6*exp(tf)+235/54*exp(3*tf)-29/6*exp( ...
5*tf)+3/2*exp(7*tf))/((3+I)*exp((1+3*i)*tf)-4/3-i-5/3*exp(6*i*tf)),tf) + CONSTANT)* ...
exp(-10*int((2*cos(3*tf)^2*exp(tf)+(3*exp(2*tf)-3)*sin(3*tf)-2*exp(tf))/(2*cos( ...
3*tf)^2*exp(tf)+(-6*sin(3*tf)*exp(tf)+9*exp(2*tf)-9)*cos(3*tf)+(9*exp(2*tf)-3)* ...
sin(3*tf)+7*exp(tf)-9*exp(3*tf)),tf))
vpasolve() cannot deal with the system because it vpasolve() cannot deal with differential equations.
If you had one more boundary condition then you could do numeric simultations using ode45() or related (looks like it might be stiff, so perhaps ode23s). In that regard, see odeFunction()
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