# Cannot find solution Of 4th order ODE using dsolve: Warning: Unable to find explicit solution.

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Dhrumin Bhavsar on 13 Aug 2019
Answered: Jyothis Gireesh on 19 Aug 2019
Below is the Code I used to solve 4th order differential equation
but I am gitting warning regarding
clc;
clear all;
syms x v(x) a0 a1 a2 a3 a4 L I E q
assume(L>0)
assume(E>0)
assume(I>0)
ode = I*E*diff(v,x,4) - q == 0
v(x) = a0+ a1*x + a2*(x^2) + a3*(x^3) + a4*(x^4)
Dv = diff(v,x);
D2v = diff(v,x,2);
D3v = diff(v,x,3);
cond1 = v(0) == 0
cond2 = Dv(0) == 0
cond3 = D2v(L) == 0,a2
cond4 = D3v(L) == 0,a3
conds = [cond1 cond2 cond3 cond4]
tSol(x) = dsolve(ode,conds)
Outpt:
Warning: Unable to find explicit solution.
ode(x) =
v(x) =
cond1 =
cond2 =
cond3 =
a2 =
cond4 =
a3 =
conds =
Warning: Unable to find explicit solution.
tSol(x) =
[ empty sym ]
Star Strider on 13 Aug 2019
The problem is most likely your initial conditions, since:
Vsol = dsolve(ode)
produces:
Vsol =
(q*x^4)/(24*E*I) + (C8*x^3)/6 + (C9*x^2)/2 + C10*x + C11
Consider that:
L = solve(cond3, L)
produces:
Warning: Solutions are valid under the following conditions: (3*a3 + 3^(1/2)*(3*a3^2 - 8*a2*a4)^(1/2))/a4 < 0;
(3*a3 - 3^(1/2)*(3*a3^2 - 8*a2*a4)^(1/2))/a4 < 0. To include parameters and conditions in the solution, specify the
'ReturnConditions' value as 'true'.
> In solve>warnIfParams (line 482)
In solve (line 357)
L =
-(3*a3 + 3^(1/2)*(3*a3^2 - 8*a2*a4)^(1/2))/(12*a4)
-(3*a3 - 3^(1/2)*(3*a3^2 - 8*a2*a4)^(1/2))/(12*a4)

Jyothis Gireesh on 19 Aug 2019
I am assuming here that you want to form an analytical solution to the given 4th order ODE. It may not be considered a good practice to provide a closed form solution for v(x) prior to calling dsolve(). This may also interfere with the solver thereby raising an ‘no explicit solution’ warning.
You may make use of the modified code attached below
clc;
clear;
syms x v(x) L I E q
assume(L>0)
assume(E>0)
assume(I>0)
ode = I*E*diff(v,x,4) - q == 0;
Dv = diff(v,x);
D2v = diff(v,x,2);
D3v = diff(v,x,3);
cond1 = v(0) == 0;
cond2 = Dv(0) == 0;
cond3 = D2v(L) == 0;
cond4 = D3v(L) == 0;

conds = [cond1 cond2 cond3 cond4];
tSol(x) = dsolve(ode,conds)
I obtained the solution as
tSol(x) =
(q*L^2*x^2)/(4*E*I) - (q*L*x^3)/(6*E*I) + (q*x^4)/(24*E*I)
You may also make use of the following documentation if you need any further clarification about the same.

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