# [✓] Solve a system of DEs with initial conditions?

Posted 8 months ago
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 So I need to solve a system of 3 DEs, I have 3 unknown functions and 4 initial conditions which is enough to find particular solutions.I have tried doing it myself and I came up with a solution: Then I tried using the DSolve[] function but it keeps returning an error. What's the problem? How do I do it?
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Posted 8 months ago
 In[1]:= Needs["VariationalMethods"] L = z'[x]^2 - 2*(x*Cos[x] + Sin[x])*z'[x] - y[x]^2 + l[x]*(z[x] - y'[x] - x*Sin[x]); eq1 = EulerEquations[L, y[x], x]; eq2 = EulerEquations[L, z[x], x]; eq3 = {-x*Sin[x] + z[x] - y'[x] == 0, y[0] == 1, y[Pi/2] == 0, z[0] == 0, z[Pi/2] == -1}; In[6]:= s = DSolve[{eq1, eq2, eq3}, {l, y, z}, x] Out[6]= {{l -> Function[{x}, -(1/8) E^(-(\[Pi]/2) - x) (-2 E^(\[Pi]/2) \[Pi] + 2 E^\[Pi] \[Pi] + 2 E^(2 x) \[Pi] + 2 E^(\[Pi]/2 + 2 x) \[Pi] - 4 E^(\[Pi]/2 + x) Cos[x] + 2 E^x \[Pi] Cos[x] + 2 E^(\[Pi] + x) \[Pi] Cos[x] + 4 E^(\[Pi]/2 + x) Cos[x] Cos[2 x] - 16 E^(\[Pi]/2 + x) Sin[x] - 2 E^x \[Pi] Sin[x] - 4 E^(\[Pi]/2 + x) \[Pi] Sin[x] + 2 E^(\[Pi] + x) \[Pi] Sin[x] - 2 E^(\[Pi]/2) Cos[x] Sin[x] + 2 E^(\[Pi]/2 + 2 x) Cos[x] Sin[x] + 4 E^(\[Pi]/2 + x) Cos[x] Sin[x]^2 + E^(\[Pi]/2) Sin[2 x] - E^(\[Pi]/2 + 2 x) Sin[2 x] + 2 E^(\[Pi]/2 + x) Sin[x] Sin[2 x])], y -> Function[{x}, -(1/16) E^(-(\[Pi]/2) - x) (2 E^(\[Pi]/2) \[Pi] - 2 E^\[Pi] \[Pi] + 2 E^(2 x) \[Pi] + 2 E^(\[Pi]/2 + 2 x) \[Pi] - 16 E^(\[Pi]/2 + x) Cos[x] - 2 E^x \[Pi] Cos[x] - 4 E^(\[Pi]/2 + x) \[Pi] Cos[x] + 2 E^(\[Pi] + x) \[Pi] Cos[x] - 4 E^(\[Pi]/2 + x) Sin[x] - 2 E^x \[Pi] Sin[x] - 2 E^(\[Pi] + x) \[Pi] Sin[x] + 2 E^(\[Pi]/2) Cos[x] Sin[x] + 2 E^(\[Pi]/2 + 2 x) Cos[x] Sin[x] + 4 E^(\[Pi]/2 + x) Cos[x]^2 Sin[x] - 4 E^(\[Pi]/2 + x) Cos[2 x] Sin[x] - E^(\[Pi]/2) Sin[2 x] - E^(\[Pi]/2 + 2 x) Sin[2 x] + 2 E^(\[Pi]/2 + x) Cos[x] Sin[2 x])], z -> Function[{x}, -(1/16) E^(-(\[Pi]/2) - x) (-2 E^(\[Pi]/2) \[Pi] + 2 E^\[Pi] \[Pi] + 2 E^(2 x) \[Pi] + 2 E^(\[Pi]/2 + 2 x) \[Pi] + 4 E^(\[Pi]/2 + x) Cos[x] - 2 E^x \[Pi] Cos[x] - 2 E^(\[Pi] + x) \[Pi] Cos[x] - 4 E^(\[Pi]/2 + x) Cos[x] Cos[2 x] + 16 E^(\[Pi]/2 + x) Sin[x] + 2 E^x \[Pi] Sin[x] + 4 E^(\[Pi]/2 + x) \[Pi] Sin[x] - 2 E^(\[Pi] + x) \[Pi] Sin[x] - 16 E^(\[Pi]/2 + x) x Sin[x] - 2 E^(\[Pi]/2) Cos[x] Sin[x] + 2 E^(\[Pi]/2 + 2 x) Cos[x] Sin[x] - 4 E^(\[Pi]/2 + x) Cos[x] Sin[x]^2 + E^(\[Pi]/2) Sin[2 x] - E^(\[Pi]/2 + 2 x) Sin[2 x] - 2 E^(\[Pi]/2 + x) Sin[x] Sin[2 x])]}} `