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l tried it with Solve,DSolve and NDSolve functions,but all failed.Should l simplify the equations in the papers or use other mathematica functions.Thanks~

  • $\begingroup$ Welcome to Mathematica.SE! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Read the faq! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! $\endgroup$ – Michael E2 Feb 3 '15 at 3:08
  • $\begingroup$ People here generally like users to post code as Mathematica code instead of images, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. $\endgroup$ – Michael E2 Feb 3 '15 at 3:09
  • $\begingroup$ An equation is entered using == instead = (Equal instead Set). (This is not enough to solve your problem, btw). $\endgroup$ – Michael E2 Feb 3 '15 at 3:10

My vnis are the integrals of your vns

eq1 = 1/2 - Integrate[Sin[v1i[a]] Cos[ArcTan[v1i'[a]/v2i'[a]]], {a, 0, t}] == v1i'[t];
eq2 = Sqrt[3]/2 - Integrate[Sin[v1i[a]] Sin[ArcTan[v1i'[a]/v2i'[a]]], {a, 0, t}] + 5 t == v2i'[t];
eq1d = D[eq1, t];
eq2d = D[eq2, t];

sol = NDSolve[{eq1d, eq2d, v1i[0] == 0, v1i'[0] == 1, v2i[0] == 0, v2i'[0] == 1}, 
              {v1i[t], v2i[t]}, {t, 0, 1}]

Plot[{v1i[t] /. sol, v2i[t] /. sol}, {t, 0, 1}]

Mathematica graphics

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