Unable to find explicit solution

Hi everyone. I have a partial differential equation which should be equal to zero.

The code is:

syms x T

p0=1;
p1=(5*(1-x)*p0)/(5-2*x);
p2=(20*(1-x)*p0)/(5-2*x);
p3=(5*x*p0)/(5-2*x);
p4=(10*x*p0)/(5-2*x);

k1=3.87*(10^2)*exp(-(22.8*10^3)/(8.3*T));
k2=1.16*(10^8)*exp(-(114*10^3)/(8.3*T));
K1=137*(T^3.998)*exp(158.7/(8.3*T));
K2=2.5*(10^(-5))*exp(-(-32.3*10^3)/(8.3*T));
K3=5.51*(10^7)*exp(-(77.6*10^3)/(8.3*T));

r1=K2*p1*(p2^0.5);
r2=(1+K2*p1)^2;
r3=K3*(p3^2)*p4;
r4=(1+K3*p4)^2;
r5=1-(p3*(p4^2))/(p1*(p2^4)*K1);
r=((k1*r1/r2)-(k2*r3/r4))*r5;

dfT=diff(r,T);
eqn=dfT==0;
solve=solve(eqn,x);

I tried to get x in terms of T. However, the result showed 'unable to find explicit solution'

Warning: Unable to find explicit solution. For options, see help. 
> In sym/solve (line 317)

I gave x an value, tring to get a set of [x,T], but the same result

dfT=diff(r,T);
dfT1=subs(dfT,x,0.5);
eqn=dfT1==0;
solve=solve(eqn,T);

Warning: Unable to find explicit solution. For options, see help. 
> In sym/solve (line 317)

What's wrong with this?

Answer 1

[Walter Roberson]

That equation is much too complex to hope for a closed form solution.

You will not be able to find a symbolic solution.

For any given T value, you might be able to find numeric solutions, possibly.

[Yuang Zhang]

Thanks!

I tried to give x and T values to get the trend of r, and found r had extreme point (maximum). But in differential equation, the result contains imaginary number. Is this wrong?

[Walter Roberson]

K1=137*(T^3.998)*exp(158.7/(8.3*T));

That is going to have imaginary results for negative T.