# Bernoulli ODE Solver API

> For the complete documentation index, see [llms.txt](/llms.txt).

Solve y' + p·y = q·y^n by the v = y^(1-n) substitution, with the linearised form and an RK4 cross-check.

Use this endpoint to add the tool to your app, site, workflow, or agent without rebuilding formulas, validation, examples, and docs.

- Slug: `bernoulli-ode-solver`
- Category: `math`
- Version: `2026-04-22`
- Credits per successful call: 1
- Starts on: free
- Max payload: 65536 bytes
- Privacy mode: `hash_only`
- Available on: free, starter, pro, business, scale
- Browser tool: https://miniwebtool.com/bernoulli-ode-solver/

## Endpoint

`POST https://api.miniwebtool.com/v1/tools/bernoulli-ode-solver/run`

## Request body

| Field | Type | Required | Description |
| --- | --- | --- | --- |
| `p` | `float` | no | constant coefficient p in y' + p·y = q·y^n |
| `q` | `float` | no | constant coefficient q |
| `n` | `float` | no | exponent n (n != 1) |
| `x0` | `float` | no | initial x |
| `y0` | `float` | no | initial y (positive) |
| `x_end` | `float` | no | x to evaluate at |

## Example request

```json
{
  "p": 1.0,
  "q": 1.0,
  "n": 2.0,
  "x0": 0.0,
  "y0": 0.5,
  "x_end": 1.0
}
```

## cURL

```bash
curl -X POST https://api.miniwebtool.com/v1/tools/bernoulli-ode-solver/run \
  -H 'Authorization: Bearer mwt_live_YOUR_KEY' \
  -H 'Content-Type: application/json' \
  -d '{"p":1.0,"q":1.0,"n":2.0,"x0":0.0,"y0":0.5,"x_end":1.0}'
```

## Response envelope

```json
{
  "request_id": "01K...",
  "tool": "bernoulli-ode-solver",
  "tool_version": "2026-04-22",
  "credits_used": 1,
  "result": {}
}
```

## Errors

Errors are RFC 7807 `application/problem+json`. Branch on `code`, not `title`.

## MCP

Agents can discover and call this tool through `https://api.miniwebtool.com/mcp` or `https://api.miniwebtool.com/v1/mcp` using MCP `tools/list` and `tools/call`.

## Frequently asked

### What does the Bernoulli ODE Solver API do?

Solve y' + p·y = q·y^n by the v = y^(1-n) substitution, with the linearised form and an RK4 cross-check. Use it to add this utility without rebuilding formulas, validation, examples, and documentation.

### How much does one call cost?

1 credit per successful call. Failed calls do not bill.

### What parameters are required?

Required fields: (none). Full schema at `https://api.miniwebtool.com/v1/openapi.json`.

### Is this deterministic?

Yes. Same input, same output. Tool version: 2026-04-22.