The file SecondOrderODES.mth defines functions for finding exact solutions of second order ordinary differential equations.  The function definitions in the file are automatically loaded when any of its functions are first used.


Sometimes more than one method may be applicable.  The most easily applied methods are described first, beginning with direct methods that produce an explicit solution and only require recognizing a pattern.  In contrast, later methods generally give implicit solutions or require more steps, such as solving a sequence of two first order differential equations.  The file FirstOrderODES.mthOA7SLO can be used to help solve these first order equations.


The notation used here and in the file SecondOrderODES.mth is the same as that described near the beginning of the help for FirstOrderODES.mthOA7SLO.  Also y=y2 at x=x2 is used as a second boundary condition, and y'=v0 at x=x0 is used as a second initial condition.  d²y/dx² is abbreviated as y" for descriptive purposes only.  In Derive the double-quote mark can not be used to denote second derivatives.


DSOLVE2(p, q, r, x, c1, c2) simplifies to an explicit general solution of the linear second order ordinary differential equation

y" + p(x)·y' + q(x)·y = r(x)

in terms of arbitrary constants c1 and c2.  Note that the last two arguments can be omitted if they are variables and you are satisfied with the names c1 and c2.


If no method applies or the equation cannot be converted to an equivalent one having a p and a q that are independent of x, DSOLVE2 returns the word "inapplicable".


DSOLVE2 can easily find a solution if p and q are numeric constants.  When q is a symbolic constant, the result will often have a more appropriate form (sinusoidal versus exponential) if q has been declared positive or negative.


A solution may contain indefinite integrals involving r(x) for which Derive can not determine a closed-form antiderivative.  To verify such a result, substitute the solution for y into the unsimplified form

⎡d ⎤2         d                    
⎢——⎥ y + p(x)·—— y + q(x)·y - r(x) 
⎣dx⎦          dx                   

and then see that it simplifies to 0.


If you want to find a specific solution for symbolic or numeric initial or boundary conditions, it is best to use DSOLVE2_IV or DSOLVE2_BV that directly return specific solutions.  Otherwise, after finding a general solution using DSOLVE2, you must substitute these constraints into general solution, solve for c1 and c2, and then substitute those values back into the general solution.


DSOLVE2_BV(p, q, r, x, x0, y0, x2, y2) is similar to DSOLVE2, but simplifies to a specific solution that satisfies the boundary conditions y=y0 at x=x0 and y=y2 at x=x2.


DSOLVE2_IV(p, q, r, x, x0, y0, v0) is similar toDSOLVE2_BV, but simplifies to a specific solution that satisfies the initial conditions y=y0 and y'=v0 at x=x0.


AUTONOMOUS_CONSERVATIVE(q, x, y, x0, y0, v0) simplifies to an implicit algebraic solution of an equation of the form y"=q(y) having initial conditions y=y0 and y'=v0 at x=x0.  Equations of this form are autonomous because the variable x does not occur and conservative because y' does not occur.  If instead of the initial condition y'=v0 you have a second boundary condition y=y2 at x=x2, substitute x2 for x and y2 for y in the solution and solve for v0, which you can then eliminate in favor of x2 and y2.


LIOUVILLE(p, q, x, y, c1, c2) simplifies to an implicit algebraic solution of the Liouville differential equation

y" + p(x)·y' + q(y)·(y')² = 0


This solution is a linear combination of two arbitrary constants c1 and c2.  Consequently, you can substitute two sets of initial or boundary conditions, and then solve two simultaneous linear equations for c1 and c2.


To solve an equation missing the dependent variable y:  f(y", y', x) = 0, use a new variable v for y' and use v' for y" to obtain a first order equation for v.  If you can solve that first order equation to obtain an algebraic equation relating v, x, x0, and v0, then noting that v=y', you can attempt to solve this subsequent first order equation to obtain an algebraic equation relating y, x, x0, y0, and v0.  The file FirstOrderODES.mthOA7SLO may help solve the two successive first order equations.


AUTONOMOUS(r, v) simplifies an expression for dv/dy, given an autonomous equation y" = r(y, v) with v representing y'.  Equations of this form are autonomous because the independent variable x doesn't occur.  If you can solve this resulting first order equation to obtain an algebraic equation relating v, y, v0, and y0, then noting that v=y', this result is another first order differential equation, which you can then attempt to solve to obtain an algebraic equation relating y, x, x0, v0, and y0.  The file FirstOrderODES.mthOA7SLO may help solve the two successive first order equations.  If r is also independent of v, instead use the function AUTONOMOUS_CONSERVATIVE to solve the equation in one step.


EXACT2(p, q, x, y, v, c) simplifies to an algebraic equation relating the solution of the equation relating y, v, x, and an arbitrary symbolic constant c, provided the equation

p(x, y, v)·y" + q(x, y, v) = 0

is exact.  Note that here the variable v is used to represent y'.  You can then attempt to solve the resulting first order differential equation.  If the equation is inexact, EXACT2 returns "inapplicable".


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