symbolic_ode.hpp

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/**
 * @file symbolic_ode.hpp
 * @brief 常微分方程求解:可分离变量,一阶线性,二阶常系数.
 */
#pragma once
#include "computation_context.hpp"
#include "symbolic.hpp"
#include <memory>
#include <string>
 
namespace LMCAS {
 
// Forward declaration to avoid circular include.
class AssumptionContext;
 
/**
 * @brief 求解可分离变量型 ODE:dy/dx = rhs(x, y)
 * @param rhs 方程右端表达式
 * @param x 自变量名
 * @param y 因变量名
 * @param ctx Optional assumption context. When y is known Positive, preserves
 *            the positive-branch marker by returning abs(F). nullptr and the
 *            default argument have identical behavior.
 * @return For rhs = f(x)*g(y), returns the first integral
 *         F(x,y) = integral(1/g(y), y) - integral(f(x), x), with implicit
 *         solutions F = C, not an explicit y(x). This describes regular regions
 *         of the original ODE where g is finite and nonzero and the selected
 *         primitives are differentiable (logarithms use their chosen branch).
 *         Equilibria at zeros of g and singular points are not enumerated.
 *         For a y-independent rhs, g = 1; in particular rhs = 0 returns y.
 *         The abs(F) marker is differentiable only away from F = 0.
 *         Unsupported primitives may remain as symbolic integrals. Returns
 *         nullptr for invalid input or a rhs not recognized by is_separable.
 */
LMCAS_API std::shared_ptr<SymbolicExpr> solve_separable_ode(
    std::shared_ptr<SymbolicExpr> rhs,
    const std::string& x,
    const std::string& y,
    const AssumptionContext* ctx = nullptr
);
 
/**
 * @brief 求解一阶线性 ODE:dy/dx + P(x)*y = Q(x)
 * @param Px 系数函数 P(x)
 * @param Qx 非齐次项 Q(x)
 * @param x 自变量名
 * @param y 因变量名
 * @param ctx Optional assumption context. When provided and the dependent
 *            variable y is known Positive, the solver prefers positive solution
 *            branches. When a coefficient is known NonZero, the solver skips
 *            zero-coefficient degenerate case checks. When nullptr, behavior is
 *            identical to the unparameterized call.
 * @return 通解的符号表达式
 */
LMCAS_API std::shared_ptr<SymbolicExpr> solve_linear1_ode(
    std::shared_ptr<SymbolicExpr> Px,
    std::shared_ptr<SymbolicExpr> Qx,
    const std::string& x,
    const std::string& y,
    const AssumptionContext* ctx = nullptr
);
 
/**
 * @brief 求解二阶常系数线性 ODE:a*y'' + b*y' + c*y = f(x)
 *
 * 齐次二阶路径与高阶 ODE 共用经过系数归一化、稳定二次公式和后向误差
 * 验证的特征根实现；统一乘以任意有限非零常数不改变解。
 * @param a 二阶导数系数
 * @param b 一阶导数系数
 * @param c 零阶项系数
 * @param fx 非齐次项 f(x)
 * @param x 自变量名
 * @param y 因变量名
 * @param ctx Optional assumption context. When the dependent variable y is
 *            known Positive, the solver prefers positive solution branches.
 * @return 通解的符号表达式
 */
LMCAS_API std::shared_ptr<SymbolicExpr> solve_linear2_ode(
    double a, double b, double c,
    std::shared_ptr<SymbolicExpr> fx,
    const std::string& x,
    const std::string& y,
    const AssumptionContext* ctx = nullptr
);
 
/**
 * @brief 求解二阶常系数常微分方程并显式报告错误.
 *
 * 当前支持齐次方程,以及 a == 0 且 b != 0 的一阶退化情形。
 * 二阶非齐次方程通过 `CasErrc::Inconclusive` 表示支持域边界；
 * 非有限系数及无法表示的特征根不会生成含 NaN/Inf 的伪解。
 */
LMCAS_API Result<std::shared_ptr<SymbolicExpr>> solve_linear2_ode_checked(
    double a, double b, double c,
    std::shared_ptr<SymbolicExpr> fx,
    const std::string& x,
    const std::string& y,
    ComputationContext& context,
    const AssumptionContext* ctx = nullptr
);
 
LMCAS_API Result<std::shared_ptr<SymbolicExpr>> solve_linear2_ode_checked(
    double a, double b, double c,
    std::shared_ptr<SymbolicExpr> fx,
    const std::string& x,
    const std::string& y,
    const AssumptionContext* ctx = nullptr
);
 
}