> ## Documentation Index > Fetch the complete documentation index at: https://docs.fincept.in/llms.txt > Use this file to discover all available pages before exploring further. # Solve Ordinary Differential Equation (ODE) > Solve an initial value problem (IVP) for a system of ordinary differential equations: dy/dt = f(t, y) with y(t0) = y0. Supports built-in systems: exponential_decay (population decay, radioactive decay), harmonic (spring oscillations), lotka_volterra (predator-prey dynamics), and van_der_pol (nonlinear oscillator). Methods: euler (simple, first-order), rk4 (Runge-Kutta 4th order, classic), rk45 (adaptive Runge-Kutta with error control). Essential for modeling dynamics, term structure models, and time-dependent processes. [Tier: BASIC, Credits: 1] ## OpenAPI ````yaml api-specs/numerical.json post /quantlib/numerical/ode/solve openapi: 3.1.0 info: title: FinceptQuantLib API - Numerical description: >- Numerical module endpoints for FinceptQuantLib API. The Numerical module (Basic Tier, 1 credit per request) provides comprehensive numerical methods including finite difference differentiation, Fast Fourier Transform (FFT), numerical integration (quadrature and Monte Carlo), interpolation methods (linear, cubic, spline), linear algebra operations (matrix decomposition, solving systems), ODE solvers, root finding algorithms, optimization methods, and nonlinear least squares fitting. Essential for quantitative analysis, scientific computing, and numerical model implementation. version: 3.0.0 contact: name: Fincept API Support url: https://fincept.in servers: - url: https://api.fincept.in description: Fincept API Production Server security: - APIKeyHeader: [] tags: - name: quantlib-numerical description: >- Numerical methods including differentiation, FFT, integration, interpolation, linear algebra, ODE solvers, root finding, and optimization x-displayName: Numerical paths: /quantlib/numerical/ode/solve: post: tags: - quantlib-numerical summary: Solve Ordinary Differential Equation (ODE) description: >- Solve an initial value problem (IVP) for a system of ordinary differential equations: dy/dt = f(t, y) with y(t0) = y0. Supports built-in systems: exponential_decay (population decay, radioactive decay), harmonic (spring oscillations), lotka_volterra (predator-prey dynamics), and van_der_pol (nonlinear oscillator). Methods: euler (simple, first-order), rk4 (Runge-Kutta 4th order, classic), rk45 (adaptive Runge-Kutta with error control). Essential for modeling dynamics, term structure models, and time-dependent processes. [Tier: BASIC, Credits: 1] operationId: solve_ode requestBody: required: true content: application/json: schema: type: object required: - system - t_span - y0 properties: system: type: string description: ODE system to solve enum: - exponential_decay - harmonic - lotka_volterra - van_der_pol example: harmonic t_span: type: array items: type: number minItems: 2 maxItems: 2 description: Time span [t0, tf] for integration example: - 0 - 10 y0: type: array items: type: number description: >- Initial conditions y(t0) - number of elements depends on system (1 for exponential_decay, 2 for harmonic/van_der_pol/lotka_volterra) example: - 1 - 0 method: type: string description: Integration method enum: - euler - rk4 - rk45 default: rk45 example: rk45 h: type: number description: Step size (for euler and rk4 methods) default: 0.01 example: 0.01 params: type: array items: type: number description: >- System-specific parameters: exponential_decay[k], harmonic[omega], lotka_volterra[a,b,c,d], van_der_pol[mu] example: - 1 example: system: harmonic t_span: - 0 - 10 y0: - 1 - 0 method: rk45 h: 0.01 params: - 1 responses: '200': description: Successful Response content: application/json: schema: type: object properties: success: type: boolean example: true data: type: object properties: system: type: string description: System that was solved example: harmonic method: type: string description: Method used example: rk45 t: type: array items: type: number description: Time points where solution was computed example: - 0 - 0.1 - 0.2 - 0.3 'y': type: array items: type: array items: type: number description: >- Solution values at each time point (each row is state at time t[i]) example: - - 1 - 0 - - 0.995 - -0.0998 - - 0.98 - -0.199 - - 0.955 - -0.296 n_steps: type: integer description: Number of time steps computed example: 100 example: success: true data: system: harmonic method: rk45 t: - 0 - 0.1 - 0.2 - 0.3 'y': - - 1 - 0 - - 0.995 - -0.0998 - - 0.98 - -0.199 - - 0.955 - -0.296 n_steps: 100 '401': $ref: '#/components/responses/UnauthorizedError' '402': $ref: '#/components/responses/InsufficientTierError' '422': description: Validation Error security: - APIKeyHeader: [] components: responses: UnauthorizedError: description: Authentication information is missing or invalid content: application/json: schema: type: object properties: detail: type: string example: Invalid API key InsufficientTierError: description: API tier insufficient for this endpoint content: application/json: schema: type: object properties: detail: type: string example: Endpoint requires Basic tier or higher securitySchemes: APIKeyHeader: type: apiKey in: header name: X-API-Key description: >- API key for authentication. Get your key at https://api.fincept.in/auth/register ````