nonlinear system problem (DSolve syntax)

Questions
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1

I need to resolve this system of equation

v1[x_]''''=q/B
v2[x_]''''+t2^2*v2[x_]''=0
v3[x_]''''+t3^2*v3[x_]''=0

then:

 v1[x_] := A1 + B1*x + C1*x^2 + D1*x^3 + q*x^4/(24*B)
 v2[x_] := A2 + B2*x + C2*Cos[t2*x] + D2*Sin[t2*x] 
 v3[x_] := A3 + B3*x + C3*Cos[t3*x] + D3*Sin[t3*x]

with B>0 and p2 and p3 are in function of q

t2 = Sqrt[p2/B]
t3 = Sqrt[p3/B]

I have 14 boundary condition

eq1 := v1[0] == 0
eq2 := v1''[0] == 0
eq3 := v3[0] == 0
eq4 := v3''[0] == 0
eq5 := v2'[l/2] == 0
eq6 := - v2'''[l/2] - t2^2 v2'[l/2] == 0
eq7 := v1'[l/2] - v3'[l] == 0
eq8 := v2'[0] - v3'[l] == 0
eq9 := - v1''[l/2] + v2''[0] - v3''[l] == 0
eq10 := v1[l/2] == 0
eq11 := v2[0] == 0
eq12 := v3[l] == 0
eq13 := -t2^2 - (- v3'''[l] - t3^2 v3'[l]) == 0
eq14 := -v1'''[l/2] - (-v2'''[0] - t2^2 v2'[0]) + t3^2 == 0

for the 14 unknowns constant (A1,B1,C1,D1,A2,B2,C2,D2,A3,B3,C3,D3,p2,p3). I need to resolve the nonlinear system, find p2 and p3, put them into the system to find q by doing the determinant of the matrix of coefficient=0.

my problem is that i don’t know how to resolve the nonlinear system

2

What does this mean?

v1[x_]''''=q/B
v2[x_]''''+t2^2*v2[x_]''=0
v3[x_]''''+t3^2*v3[x_]''=0
3

I have to resolve a frame composed by 3 beams. these 3 differential equations are the elastic lines of the 3 beams. v2 and v3 are the elastic lines using a second order analysis. I need to find the critical load q.

4

Sorry Diana-- I didn’t recognize the syntax. If these are differential equations the Mathematica syntax is

v1''''[x] == q/B
v2''''[x] + t2^2*v2''[x] == 0
v3''''[x] + t3^2*v3[x] == 0

Note the double equal for equality rather than assignment..

You might have a look at this:

How to work with differential equations

Kind regards,

David

5

oh yes, sorry, I’m wrong, today I’ve worked on and I found this nonlinear system

> sys:={D2 t2 Cos[(l t2)/2] == C2 t2 Sin[(l t2)/2], 
         B1 + (3 D1 l^2)/4 + (l^3 q)/(48 B) == B3 + D3 t3 Cos[l t3], 
         D2 t2 == B3 + D3 t3 Cos[l t3], 
         t3^2 D3 Sin[l t3] == 3 D1 l + (l^2 q)/(8 B) + C2 t2^2, 
         48 l (4 B1 + l (D1 l)) + (l^4 q)/B == 0, 
         A2 + C2 == 0, 
         B3 l + D3 Sin[l t3] == 0, 
         t2^2 == B3 t3^2, 
         12 D1 + (l q)/B == 2 t3^2}

I tried to do

 Solve[sys,{B1, D1, A2, C2, D2, B3, D3, t2, t3}]
    DSolve[sys,{B1, D1, A2, C2, D2, B3, D3, t2, t3}]

but it doesn’t work…how can I solve it?