# Formatting the results from solving a differential equation

{{x[t] -> 600 (t + I π t - 50 Log[50] + t Log[50] + 50 Log[50 - t] - t Log[-50 + t])}}


After solving a differential equation, I get an answer in a form where I want to get rid of irrational values. How can it be implemented? Function N doesn't help.

x0 = 15000; y0 = 10; m0 = 150; v0 = 210; G = 3; u = 600; g = 9.8;
v1[h_] := Sqrt[v0^2 - 2 g h];
t1[h_] := (v0 - v1[h])/g;
M[t_] := m0 - G t;
DSolve[{M[t] x ''[t] == G u, x[0] == 0, x'[0] == 0}, x[t], t] // FullSimplify
DSolve[{M[t] y ''[t] == -M[t] g, y[0] == h, y'[0] == v1[h]}, y[t], t] // FullSimplify


This is the full code fragment. I need to obtain a plot.

• What was the equation? What irrational values do you mean? As posted your question is too vague to answer. Jun 1, 2015 at 14:08
• In my equation was iPit, moderator edited my post addition of the tag code. I need to get rid of i (root of minus one).
– Mike
Jun 1, 2015 at 16:55

## 2 Answers

EDIT: added detail for provided equations.

$Version  "10.1.0 for Mac OS X x86 (64-bit) (March 24, 2015)" x0 = 15000; y0 = 10; m0 = 150; v0 = 210; G = 3; u = 600; g = 98/10; v1[h_] = Sqrt[v0^2 - 2 g h]; t1[h_] = (v0 - v1[h])/g; M[t_] = m0 - G t; Clear[x, y] x[t_] = x[t] /. DSolve[ {M[t] x''[t] == G u, x[0] == 0, x'[0] == 0}, x[t], t][[1]] // Simplify  This satisfies the equations in the DSolve: {M[t] x''[t] == G u, x[0] == 0, x'[0] == 0} // Simplify  {True, True, True} To identify the conditions for x[t] to be real, Reduce[{Element[x[t], Reals], Element[t, Reals]}, t] // Quiet  t < 50 x[t_] = x[t] // FullSimplify[#, t < 50] &  Confirming the condition for x[t] real: for the Log to produce a real value, its argument must be positive, i.e., Reduce[-50/(t - 50) > 0, t]  t < 50 y[t_, h_] = y[t] /. DSolve[ {M[t] y''[t] == -M[t] g, y[0] == h, y'[0] == v1[h]}, y[t], t][[1]] // Simplify  In the equations in the DSolve the dependence on h is implicit. To substitute y[t, h] into the equations, define y[t_] = y[t, h];  This satisfies the equations in the DSolve: {M[t] y''[t] == -M[t] g, y[0] == h, y'[0] == v1[h]} // Simplify  {True, True, True} To keep y[t, h] real, the argument of the Sqrt must be nonnegative. Reduce[900 - 2 h/5 >= 0, h]  h <= 2250 Manipulate[ Column[{Plot[{x[t], y[t, h]}, {t, 0, 300/7}, AxesLabel -> {"t", "x(t),\ny(t, h)"}, PlotRange -> {{0, 300/7}, {0, 5000}}, PlotLegends -> {"x(t)", "y(t, h)"}, ImageSize -> 324], NSolve[{y[t, h] == 0, 0 < t <= 300/7}, t][[1]]}], {{h, 0}, 0, 2250, 25, Appearance -> "Labeled"}]  Manipulate[ ParametricPlot[{x[t], y[t, h]}, {t, 0, 300/7}, PlotRange -> {{0, 17500}, {0, 2275}}, AspectRatio -> 1/GoldenRatio, AxesLabel -> {"x(t)", "y(t, h)"}], {{h, 0}, 0, 2250, 25, Appearance -> "Labeled"}]  • What does mean [[1]] in first DSolve x[t] ? – Mike Jun 2, 2015 at 16:18 • Evaluate the DSolve with and without the [[1]] and compare the results. Alternatively, select either the two left brackets or the two right brackets and press F1 for help. As for the Reduce, I used Mathematica version 10.1 on a Mac, perhaps you used a different version or a different operating system. Jun 2, 2015 at 16:47 • I don't understand( Can u say me, on understandable language, what is it? Plz – Mike Jun 2, 2015 at 17:08 • Read the documentation for Part Jun 2, 2015 at 17:12 • And this part M[t] x''[t] == G u? – Mike Jun 2, 2015 at 17:17 DSolve[{M[t] x ''[t] == G u, x[0] == 0, x'[0] == 0}, x[t], t] // FullSimplify[#, t < 50] & (* {{x[t] -> 600 (t + (-50 + t) Log[-(50/(-50 + t))])}} *)  cancels the explicit imaginary term for t < 50. The imaginary term cannot be eliminated for larger t except by applying the boundary conditions in the first ODE at some t value greater than 50 rather than at t == 0. • You don't so much need to change the boundary conditions as the parameters. The original equations have a singular point when$t = m_0/G$($t = 50$for the provided code.) I suppose there might be a solution that goes to a finite value in the neighborhood of$t = m_0/G$, in which case finely tuned boundary conditions at$t = 0\$ would do the trick. Jun 2, 2015 at 14:06
• @MichaelSeifert Perhaps, I should make clear that, so long as a singularity exists at t==50, the boundary conditions must be applied for t > 50 in order to eliminate the imaginary term there. Jun 2, 2015 at 14:10