# Solving this system of Second ODE

I’m trying to solve these questions to get a single analytic solution of x[t] satisfies all the three equations

y[t_] := 3  (x’[t]/x[t] )^2-m,                    (1)

y[t_] := -2 (x’’[t]/x[t]) - (x’[t]/x[t] )^2,        (2)

y[t_] := -3 (x’’[t]/x[t] ) -3 (x’[t]/x[t] )^2.       (3)


Where m is a constant. The step I made is simplifying by setting (1)=(2)=(3), and got a single equation:

7 (x’[t]/x[t] )^2 + 5 (x’’[t]/x[t]) -m =0


Or:

x’[t]^2+ 0.7 x’’[t] x[t]- 0.14 m =0


How to solve this equation to get an analytic solution of x[t] ? and how to determine the integration constants. The initial conditions are arbitrary, but m can't vanish.

• There is does not seem to be an analytical solution for the initial conditions you give. But you can obtain the general solution using DSolve. Are you sure the IC's are correct? Notice at $t=0$ you get $0=0.14 m$ since $x(0)=0$ which means $m=0$ Commented Aug 25, 2022 at 10:24
• Use any arbitrary initial conditions. Please see the question after the edition. Commented Aug 25, 2022 at 12:41

If we set x[0]==1, it seems that m==0 is the unique solution.

y1[t_] = 3 (x'[t]/x[t])^2 - m;
y2[t_] = -2 (x''[t]/x[t]) - (x'[t]/x[t])^2;
y3[t_] = -3 (x''[t]/x[t]) - 3 (x'[t]/x[t])^2;
eqns = Simplify[y1[t] == y2[t] == y3[t]]
sol = DSolve[{(2 x'[t]^2)/x[t] + x''[t] == 0, x[0] == 1, x'[0] == 1},
x, t]
Solve[m x[t] == (4 x'[t]^2)/x[t] + 2 x''[t] /. sol[[1]] /. t -> 0]


If we set x[0]==x0,then

Clear[eqns,sol];
y1[t_] = 3 (x'[t]/x[t])^2 - m;
y2[t_] = -2 (x''[t]/x[t]) - (x'[t]/x[t])^2;
y3[t_] = -3 (x''[t]/x[t]) - 3 (x'[t]/x[t])^2;
eqns = Simplify[y1[t] == y2[t] == y3[t]]
sol = DSolve[{eqns[[2]], x[0] == x0, x'[0] == 1}, x, t]
Solve[eqns[[1]] /. sol[[1]] /. t -> 0]


The only solution is also {{m -> 0}, {x0 -> 0}}.

• What if we consider arbitrary initial conditions, but m  can't vanish. Commented Aug 25, 2022 at 12:45