Math & Numbers

First-Order ODE Solver (Separation of Variables)

Solve separable dy/dx = g(x)·h(y) symbolically: H(y) = G(x) + C, explicit inversion, initial conditions, and partial fractions for quadratics like y(1−y).

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Quick start

Call this tool from your code in three languages.

cURL

curl -X POST 'https://api.elysiatools.com/en/api/tools/first-order-ode-solver' \
  -H 'Content-Type: application/json' \
  -d '{"gx":"2x","hy":"y","x0":0,"y0":1}'

API reference

Send a POST request with your inputs as JSON. File parameters require a separate upload first.

Endpoint

HTTP

POST https://api.elysiatools.com/en/api/tools/first-order-ode-solver

Parameters

Name	Type	Required	Description
gx	text	Yes	x-dependent factor of dy/dx = g(x)·h(y). Terms: constants, x^n, (kx+m)^n, e^(kx), sin(kx+m), cos(kx+m), ln(kx+m), sqrt(kx+m).
hy	text	Yes	y-dependent factor. Single terms (y, y^n, (ky+m)^n, e^(ky)) or a degree ≤ 2 polynomial with real roots, or c/D(y).
x0	number	No	Only used together with y₀ to pin down the constant C.
y0	number	No	Initial value. Leave empty for the general solution; provide with x₀ for the particular solution.

Response format

MCP integration

Add this tool to your Model Context Protocol server so AI agents can list and call it.

Server configuration

Add this block to your MCP client configuration:

mcp.json

{
  "mcpServers": {
    "elysiatools-first-order-ode-solver": {
      "name": "first-order-ode-solver",
      "description": "Solve separable dy/dx = g(x)·h(y) symbolically: H(y) = G(x) + C, explicit inversion, initial conditions, and partial fractions for quadratics like y(1−y).",
      "baseUrl": "https://api.elysiatools.com/mcp/sse?toolId=first-order-ode-solver",
      "command": "",
      "args": [],
      "env": {},
      "isActive": true,
      "type": "sse"
    }
  }
}

List available tools

After connecting to the SSE endpoint, list the exposed tools:

tools/list

{
  "jsonrpc": "2.0",
  "id": 1,
  "method": "tools/list"
}

Call this tool

Invoke the tool by its id, passing arguments built from its parameters:

tools/call

{
  "jsonrpc": "2.0",
  "id": 2,
  "method": "tools/call",
  "params": {
    "name": "first-order-ode-solver",
    "arguments": {
  "gx": "2x",
  "hy": "y",
  "x0": 0,
  "y0": 1
}
  }
}

Chain multiple tools in one session with a comma-separated toolId list, e.g. /mcp/sse?toolId=png-to-webp,jpg-to-webp,gif-to-webp (max 20).

Questions or issues? Contact [email protected]

Browse tools Open tool

Name

Type

Required

Description

text

Yes

x-dependent factor of dy/dx = g(x)·h(y). Terms: constants, x^n, (kx+m)^n, e^(kx), sin(kx+m), cos(kx+m), ln(kx+m), sqrt(kx+m).

text

Yes

y-dependent factor. Single terms (y, y^n, (ky+m)^n, e^(ky)) or a degree ≤ 2 polynomial with real roots, or c/D(y).

number

Only used together with y₀ to pin down the constant C.

number

Initial value. Leave empty for the general solution; provide with x₀ for the particular solution.

{ "mcpServers": { "elysiatools-first-order-ode-solver": { "name": "first-order-ode-solver", "description": "Solve separable dy/dx = g(x)·h(y) symbolically: H(y) = G(x) + C, explicit inversion, initial conditions, and partial fractions for quadratics like y(1−y).", "baseUrl": "https://api.elysiatools.com/mcp/sse?toolId=first-order-ode-solver", "command": "", "args": [], "env": {}, "isActive": true, "type": "sse" } } }