Using Mathematica to solve ODEs (part 1)
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1 Using Mathematica to solve ODEs (part ) Basic tool is DSolve Note that in DSolve the dependent variable (usually y below) must be written y[x] or y [x] (for the derivative) or y [x] (for the second derivative) etc.. You cannot write just y. Mathematica needs to know that y is dependent. Also don t forget to use a double equals sign: == First-order linear ODEs Apply to separable equation discussed in class: Sec. 8. # General solution. C[] is the integration constant. In[9]:= DSolve@y'@xD Sin@xD ã y@xd Log@y@xDD, y@xd, xd Out[9]= ::y@xd Ø C@D TanA x E >> Solution given BC: In[]:= DSolve@8y'@xD Sin@xD ã y@xd Log@y@xDD, y@pi ê D ã E<, y@xd, xd Out[]= ::y@xd Ø TanA x E >> Sec. 8. # (another example discussed in class) General linear first-order but non-separable In[]:= Out[]= DSolve@y'@xD + y@xd Cos@xD ã Sin@ xd, y@xd, xd 99y@xD Ø - + -Sin@xD C@D + Sin@xD== Note that this solution differs from that derived in class only in that the constant C[] here corresponds to (+y()) in the expresssion in lecture notes. This comparison is what Boas calls reconciling.
2 ODE.nb In[]:= Useful plots---methods are needed for HW Will use example of Sec. 8. # Family of solution curves Here is a plot of solutions for values of y(). These are refered to as a family of solution curves (depending, in this case, on parameter). Notice that the solutions are periodic with period, as makes sense from the ODE given the periodicity of Cos[x] and Sin[x]. yy@x_, y_d := Sin@xD - + H + yl Exp@-Sin@xDD In[9]:= solnplot = Plot@8yy@x, -D, yy@x, D, yy@x, D<, 8x,, 7<, PlotStyle Ø 88Thick, Red<, 8Thick, Purple<, 8Thick, Green<<D Out[9]= Slope plot Here is the slope plot, with the slope arrows normalized to unit length. Note that the normalized slope vector is given by {, y } / Sqrt[+(y )^].
3 ODE.nb In[]:= slopeplot = VectorPlot@ 8, Sin@ xd - y Cos@xD< ê Sqrt@ + HSin@ xd - y Cos@xDL^D, 8x,, 7<, 8y, -.5,.5<D Out[]= Overlaying the solutions on the slope plot shows how the slope plot allows you to visualize the solutions by following arrows (head to tail to head to tail etc.): In[5]:= Show@slopeplot, solnplotd Out[5]=
4 ODE.nb In[55]:= Stream Plot Here s an alternative, using StreamPlot[] in which Mathematica does the work of connecting the arrows head to tail and curving them so that one can see the flows (which are the solutions to the original differential equation). Note that in this case one does not need to normalize the slope vectors streamplot = StreamPlot@8, Sin@ xd - y Cos@xD<, 8x,, 7<, 8y, -.5,.5<D Out[55]= Overlaying shows that the head-to-tail arrows really do map out the solutions
5 ODE.nb 5 In[56]:= Show@streamplot, solnplotd Out[56]= Contour plot Here is another way of showing the family of solutions using a contour plot First rewrite the solution so that only the constant of integration (here y()) is on the RHS. Then put the LHS into contour plot. The labels on the lines give the values of y().
6 6 ODE.nb In[6]:= curvefamily = ContourPlot@Hy - Sin@xD + L ê Exp@-Sin@xDD -, 8x,, 7<, 8y, -, 5<, ContourLabels Ø TrueD Out[6]= Solutions and contours indeed match: In[7]:= Show@curvefamily, solnplotd Out[7]= Orthogonal curves
7 ODE.nb 7 These are curves orthogonal to the solutions at all points. In some cases one can solve analytically for these (as in the examples in Boas and in one of the HW problems), but in the present case one cannot. However one can visualize these curves using StreamPlot[] without having an explicit solution. Note that (as discussed in class lecture notes) the slope of the orthogonal curve at a given point {x,y} is given by -/(slope of solution). The directions of the arrows in the following plot should be ignored. In[68]:= orthogcurves = StreamPlot@8, - ê HSin@ xd - y Cos@xDL<, 8x,, 7<, 8y, -, 5<, StreamStyle Ø 8Red, Thick<D 5 Out[68]= Superimposing on the family of solution curves, one sees that the orthogonal trajectories can be thought of as paths of steepest descent over the landscape of the solutions.
![8 ODE.nb In[69]:= Show@curvefamily, orthogcurvesd Out[69]= Second-order, linear ODEs Sec 8.5 #7 Simple second order linear with constant coeffs and homogeneous. Answer agrees with lecture notes.](/docs-images/87/97032535/images/8-0.jpg "In[7]:= Out[7]= DSolve@y''@xD - 5 y'@xd + 6 y@xd ã, y@xd, xd 99y@xD Ø x C@D + x C@D== In[8]:= Sec. 8.6 #5 Second-order linear with constant coeffs and non-zero RHS. Answer agrees with lecture notes.")
8 8 ODE.nb In[69]:= orthogcurvesd Out[69]= Second-order, linear ODEs Sec 8.5 #7 Simple second order linear with constant coeffs and homogeneous. Answer agrees with lecture notes. In[7]:= Out[7]= - 5 y'@xd + 6 y@xd ã, y@xd, xd 99y@xD Ø x C@D + x C@D== In[8]:= Sec. 8.6 #5 Second-order linear with constant coeffs and non-zero RHS. Answer agrees with lecture notes. DSolve@y''@xD - y'@xd - y@xd ã 6 x^ Exp@-xD, y@xd, xd Out[8]= ::y@xd Ø - -x I + x + x + x M + -x C@D + x C@D>>
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