# Solving a system of linear differential equations [closed]

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:

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

## closed as off-topic by corey979, m_goldberg, C. E., march, ÖskåSep 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.
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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]}}


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


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


 DSolve[system, U[t], t]


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);


 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) }}$$

• 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 – Brad Sep 2 at 20:54