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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~

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  • $\begingroup$ 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
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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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