# ElementaryFunctionODESolver(R, F)¶

ElementaryFunctionODESolver provides the top-level functions for finding closed form solutions of ordinary differential equations and initial value problems.

solve: (Equation F, BasicOperator, Equation F, List F) -> Union(F, failed)
solve(eq, y, x = a, [y0, ..., ym]) returns either the solution of the initial value problem eq, y(a) = y0, y'(a) = y1, ... or “failed” if the solution cannot be found; error if the equation is not one linear ordinary or of the form dy/dx = f(x, y).
solve: (Equation F, BasicOperator, Symbol) -> Union(Record(particular: F, basis: List F), F, failed)
solve(eq, y, x) returns either a solution of the ordinary differential equation eq or “failed” if no non-trivial solution can be found; If the equation is linear ordinary, a solution is of the form [h, [b1, ..., bm]] where h is a particular solution and [b1, ...bm] are linearly independent solutions of the associated homogenuous equation f(x, y) = 0; A full basis for the solutions of the homogenuous equation is not always returned, only the solutions which were found; If the equation is of the form {dy/dx = f(x, y)}, a solution is of the form h(x, y) where h(x, y) = c is a first integral of the equation for any constant c; error if the equation is not one of those 2 forms.
solve: (F, BasicOperator, Equation F, List F) -> Union(F, failed)
solve(eq, y, x = a, [y0, ..., ym]) returns either the solution of the initial value problem eq, y(a) = y0, y'(a) = y1, ... or “failed” if the solution cannot be found; error if the equation is not one linear ordinary or of the form dy/dx = f(x, y).
solve: (F, BasicOperator, Symbol) -> Union(Record(particular: F, basis: List F), F, failed)
solve(eq, y, x) returns either a solution of the ordinary differential equation eq or “failed” if no non-trivial solution can be found; If the equation is linear ordinary, a solution is of the form [h, [b1, ..., bm]] where h is a particular solution and and [b1, ...bm] are linearly independent solutions of the associated homogenuous equation f(x, y) = 0; A full basis for the solutions of the homogenuous equation is not always returned, only the solutions which were found; If the equation is of the form {dy/dx = f(x, y)}, a solution is of the form h(x, y) where h(x, y) = c is a first integral of the equation for any constant c.
solve: (List Equation F, List BasicOperator, Symbol) -> Union(Record(particular: Vector F, basis: List Vector F), failed)
solve([eq_1,...,eq_n], [y_1,...,y_n], x) returns either “failed” or, if the equations form a fist order linear system, a solution of the form [y_p, [b_1, ..., b_n]] where h_p is a particular solution and [b_1, ...b_m] are linearly independent solutions of the associated homogenuous system. error if the equations do not form a first order linear system
solve: (List F, List BasicOperator, Symbol) -> Union(Record(particular: Vector F, basis: List Vector F), failed)
solve([eq_1,...,eq_n], [y_1,...,y_n], x) returns either “failed” or, if the equations form a fist order linear system, a solution of the form [y_p, [b_1, ..., b_n]] where h_p is a particular solution and [b_1, ...b_m] are linearly independent solutions of the associated homogenuous system. error if the equations do not form a first order linear system
solve: (Matrix F, Symbol) -> Union(List Vector F, failed)
solve(m, x) returns a basis for the solutions of D y = m y. x is the dependent variable.
solve: (Matrix F, Vector F, Symbol) -> Union(Record(particular: Vector F, basis: List Vector F), failed)
solve(m, v, x) returns [v_p, [v_1, ..., v_m]] such that the solutions of the system D y = m y + v are v_p + c_1 v_1 + ... + c_m v_m where the c_i's are constants, and the v_i's form a basis for the solutions of D y = m y. x is the dependent variable.