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I am trying to solve a system of linear differential equations, and I am following the instructions given on the Wolfram Alpha page.

I am not getting the desired output, as can be seen below: enter image description here

In the last line, instead of solving the equation for me, I'm just getting my input back. Where am I going wrong?

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closed as off-topic by corey979, m_goldberg, C. E., march, Öskå Sep 12 at 18:40

This question appears to be off-topic. The users who voted to close gave these specific reasons:

  • "This question arises due to a simple mistake such as a trivial syntax error, incorrect capitalization, spelling mistake, or other typographical error and is unlikely to help any future visitors, or else it is easily found in the documentation." – m_goldberg, march, Öskå
  • "This question cannot be answered without additional information. Questions on problems in code must describe the specific problem and include valid code to reproduce it. Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included." – corey979, C. E.
If this question can be reworded to fit the rules in the help center, please edit the question.

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Mathematica can not solve this coupled ODE's. Btw, you had few syntax issues. it is Cos and not cos. Same for Sin.

You also need to convert the matrix equation to separate equations. But after doing all of this, DSolve can not solve them.

ClearAll[x, t, y, u, v]
x[t_] := Sin[t]
y[t_] := Cos[t]
A = {{x'[t], y'[t]}, {y'[t], -x'[t]}}

Mathematica graphics

U[t_] = {u[t], v[t]};
system = U'[t] == A.U[t]

Mathematica graphics

 system = First@Solve[system, U'[t]] /. Rule -> Equal

Mathematica graphics

 DSolve[system, U[t], t]

Mathematica graphics

fyi, if you want solution, Maple is able to solve this. But solution are complicated. What does this model represent? Is this an actual physical system?

restart;
ode1:=diff(u(t),t)=cos(t)*u(t)-sin(t)*v(t);
ode2:=diff(v(t),t)=-sin(t)*u(t)-cos(t)*v(t);

Mathematica graphics

 dsolve([ode1,ode2],[u(t),v(t)])

$$ v \left( t \right) ={\frac {\sqrt {2\,\sin \left( t \right) -1} \left( \sqrt [4]{-\sqrt {3}\sin \left( t \right) +3\,\cos \left( t \right) +2\,\sqrt {3}}\sqrt [4]{\sin \left( t \right) -2-\sqrt {3} \cos \left( t \right) } \left( \sin \left( t \right) +i\cos \left( t \right) \right) ^{-i/2\sqrt {3}}{\it \_C2}+\sqrt [4]{-\sqrt {3}\sin \left( t \right) -3\,\cos \left( t \right) +2\,\sqrt {3}}\sqrt [4]{ \sin \left( t \right) -2+\sqrt {3}\cos \left( t \right) } \left( \sin \left( t \right) +i\cos \left( t \right) \right) ^{i/2\sqrt {3}}{ \it \_C1} \right) }{\sqrt [4]{\sin \left( t \right) -2-\sqrt {3}\cos \left( t \right) }\sqrt [4]{\sin \left( t \right) -2+\sqrt {3}\cos \left( t \right) }}} $$

and

$$ u \left( t \right) =1/4\,{\frac {-4\,\cos \left( t \right) \left( 2\, \sin \left( t \right) -1 \right) \left( \sqrt [4]{ \left( -\sin \left( t \right) +2 \right) \sqrt {3}-3\,\cos \left( t \right) } \sqrt [4]{\sin \left( t \right) -2+\sqrt {3}\cos \left( t \right) } \left( \sin \left( t \right) +i\cos \left( t \right) \right) ^{i/2 \sqrt {3}}{\it \_C1}+\sqrt [4]{\sin \left( t \right) -2-\sqrt {3}\cos \left( t \right) } \left( \sin \left( t \right) +i\cos \left( t \right) \right) ^{-i/2\sqrt {3}}\sqrt [4]{ \left( -\sin \left( t \right) +2 \right) \sqrt {3}+3\,\cos \left( t \right) }{\it \_C2} \right) \left( \sin \left( t \right) -2-\sqrt {3}\cos \left( t \right) \right) \left( \sin \left( t \right) -2+\sqrt {3}\cos \left( t \right) \right) \left( \sin \left( t \right) +i\cos \left( t \right) \right) \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}-3\,\cos \left( t \right) \right) ^{3/4} \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}+3\,\cos \left( t \right) \right) ^{3/4}-4\,{\it \_C1}\, \left( \sin \left( t \right) +i\cos \left( t \right) \right) ^{i/2\sqrt {3}} \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}-3\,\cos \left( t \right) \right) \cos \left( t \right) \left( \sin \left( t \right) -2-\sqrt {3}\cos \left( t \right) \right) \left( \sin \left( t \right) +i\cos \left( t \right) \right) \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}+3\,\cos \left( t \right) \right) ^{3/4} \left( \sin \left( t \right) -2+\sqrt {3}\cos \left( t \right) \right) ^{5/4 }-2\,i{\it \_C1}\, \left( 2\,\sin \left( t \right) -1 \right) \left( \sin \left( t \right) +i\cos \left( t \right) \right) ^{i/2\sqrt {3}} \sqrt {3} \left( \cos \left( t \right) -i\sin \left( t \right) \right) \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}-3 \,\cos \left( t \right) \right) \left( \sin \left( t \right) -2- \sqrt {3}\cos \left( t \right) \right) \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}+3\,\cos \left( t \right) \right) ^{3/4} \left( \sin \left( t \right) -2+\sqrt {3}\cos \left( t \right) \right) ^{5/4}-{\it \_C1}\, \left( 2\,\sin \left( t \right) -1 \right) \left( \sin \left( t \right) +i\cos \left( t \right) \right) ^{i/2\sqrt {3}} \left( -\sqrt {3}\cos \left( t \right) +3\, \sin \left( t \right) \right) \left( \sin \left( t \right) -2-\sqrt {3}\cos \left( t \right) \right) \left( \sin \left( t \right) +i\cos \left( t \right) \right) \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}+3\,\cos \left( t \right) \right) ^{3/4} \left( \sin \left( t \right) -2+\sqrt {3}\cos \left( t \right) \right) ^{5/4 }+{\it \_C1}\, \left( 2\,\sin \left( t \right) -1 \right) \left( \sin \left( t \right) +i\cos \left( t \right) \right) ^{i/2\sqrt {3}} \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}-3\,\cos \left( t \right) \right) \left( \cos \left( t \right) +\sqrt {3} \sin \left( t \right) \right) \left( \sin \left( t \right) +i\cos \left( t \right) \right) \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}+3\,\cos \left( t \right) \right) ^{3/4} \left( \sin \left( t \right) -2+\sqrt {3}\cos \left( t \right) \right) ^{5/4 }-4\,{\it \_C2}\, \left( \sin \left( t \right) +i\cos \left( t \right) \right) ^{-i/2\sqrt {3}} \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}+3\,\cos \left( t \right) \right) \cos \left( t \right) \left( \sin \left( t \right) -2+\sqrt {3}\cos \left( t \right) \right) \left( \sin \left( t \right) +i\cos \left( t \right) \right) \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}-3\,\cos \left( t \right) \right) ^{3/4} \left( \sin \left( t \right) -2-\sqrt {3}\cos \left( t \right) \right) ^{5/4 }+2\,i{\it \_C2}\, \left( 2\,\sin \left( t \right) -1 \right) \left( \sin \left( t \right) +i\cos \left( t \right) \right) ^{-i/2\sqrt {3} }\sqrt {3} \left( \cos \left( t \right) -i\sin \left( t \right) \right) \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}+3 \,\cos \left( t \right) \right) \left( \sin \left( t \right) -2+ \sqrt {3}\cos \left( t \right) \right) \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}-3\,\cos \left( t \right) \right) ^{3/4} \left( \sin \left( t \right) -2-\sqrt {3}\cos \left( t \right) \right) ^{5/4}-{\it \_C2}\, \left( 2\,\sin \left( t \right) -1 \right) \left( \sin \left( t \right) +i\cos \left( t \right) \right) ^{-i/2\sqrt {3}} \left( -\sqrt {3}\cos \left( t \right) -3\, \sin \left( t \right) \right) \left( \sin \left( t \right) -2+\sqrt {3}\cos \left( t \right) \right) \left( \sin \left( t \right) +i\cos \left( t \right) \right) \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}-3\,\cos \left( t \right) \right) ^{3/4} \left( \sin \left( t \right) -2-\sqrt {3}\cos \left( t \right) \right) ^{5/4 }+{\it \_C2}\, \left( 2\,\sin \left( t \right) -1 \right) \left( \sin \left( t \right) +i\cos \left( t \right) \right) ^{-i/2\sqrt {3}} \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}+3\,\cos \left( t \right) \right) \left( \cos \left( t \right) -\sqrt {3} \sin \left( t \right) \right) \left( \sin \left( t \right) +i\cos \left( t \right) \right) \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}-3\,\cos \left( t \right) \right) ^{3/4} \left( \sin \left( t \right) -2-\sqrt {3}\cos \left( t \right) \right) ^{5/4 }}{\sqrt {2\,\sin \left( t \right) -1} \left( \sin \left( t \right) +i \cos \left( t \right) \right) \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}-3\,\cos \left( t \right) \right) ^{3/4} \left( \sin \left( t \right) -2-\sqrt {3}\cos \left( t \right) \right) ^{5/4 } \left( \left( -\sin \left( t \right) +2 \right) \sqrt {3}+3\,\cos \left( t \right) \right) ^{3/4} \left( \sin \left( t \right) -2+ \sqrt {3}\cos \left( t \right) \right) ^{5/4}\sin \left( t \right) }} $$

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  • $\begingroup$ Thank you so much for the feedback!! Yes this is the trajectory of the rear end of a bicycle as the front end goes around a circle $\endgroup$ – Brad Sep 2 at 20:54

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