Notice that the general solution is a linear function of the arbitrary constant C.
This plots several integral curves of the equation for different values of a. The plot shows that the solutions have an inflection point if the parameter a lies between -1 and 1, while a global maximum or minimum arises for other values of a.
Here is the solution to a linear second-order equation with initial values prescribed for x[t] and x[t] at t0.
Here is a plot of the solutions for different initial directions. The solution approaches - or as t→- according to whether the value of x0 is less than or greater than -2, which is the largest root of the auxiliary equation for the ODE.
This problem has no solution because the term with C in the general solution vanishes at both x0 and x. Hence there are two inconsistent conditions for the parameter C and the solution is an empty set.
Since this is a fourth-order ODE, four independent conditions must be specified to find a particular solution for an IVP. If there is an insufficient number of conditions, the solution returned by DSolve may contain some of the arbitrary parameters, as follows.
The solutions x[t], y[t], and z[t] are parametrized by the variable t and can be plotted separately in the plane or as a curve in space.
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