How to solve this equation?

I have an equation that I'd like to solve for p where 0<a<1. Is there anyone who can help me with this? when I'm running my code it says "Unable to find explicit solution"

syms p a
eqn=2*(p^a)+2*(p^2)+4*(p);
s=solve(eqn,p);

[Dyuman Joshi]

You don't have an equation, you have an expression. There is no equality in your code. You need to specify the equality and even then, there is no guarantee that solve() will be able to find an explicit symbolic solution.

[Walter Roberson]

@Dyuman Joshi

solve() assumes an implict ==0 for expressions -- which also means that if you have equations of the form f(x)==g(x) then you can rewrite as f(x)-g(x)==0 and then drop the ==0 to pass in just f(x)-g(x)

Answer 1

[Walter Roberson]

There are multiple solutions. p=0 is a solution for all a values, but there are also complex solutions -- and you did not rule out complex solutions.

You are not going to be able to get a closed form solution for an equation involving sums with a value to an unknown power -- except in the limited situation where the equation happens to match a pattern that can be solved with a wrightOmega

syms p
a = sym(linspace(eps, 1-eps, 20),'f')

a = 

eqns = 2*(p.^a) + 2*(p^2) + 4*(p);
sols = arrayfun(@(E)vpasolve(E,p,-100), eqns, 'uniform', 0)

sols = 1×20 cell array
  Columns 1 through 7

    {[- 1.000000018675…]}    {[- 1.276881684015…]}    {[- 1.397919366591…]}    {[- 1.499530104971…]}    {[- 1.593743226079…]}    {[- 1.685062313890…]}    {[- 1.775741013081…]}

  Columns 8 through 14

    {[- 1.867080012385…]}    {[- 1.959877598644…]}    {[- 2.054611315759…]}    {[- 2.151509926327…]}    {[- 2.250569694613…]}    {[- 2.351534910499…]}    {[- 2.453849153663…]}

  Columns 15 through 20

    {[- 2.556577477736…]}    {[- 2.658296214598…]}    {[- 2.756945177594…]}    {[- 2.849636776102…]}    {[- 2.932419250129…]}    {[- 2.999999999999…]}

hold on;
cellfun(@(A,S) plot(repmat(A, numel(S), 1), real(S), 'k+'), num2cell(a), sols);
cellfun(@(A,S) plot(repmat(A, numel(S), 1), imag(S), 'r*'), num2cell(a), sols);
hold off

a(end),sols{end}

ans = 

ans =