- Mathematics
- Functions
- Differential Equation Solver

## Differential Equation Calculator

## Answers to Questions (FAQ)

### What is a differential equation? (Definition)

A differential equation (or diffeq) is an equation that relates an unknown function to its derivatives (of order n).

There are hom*ogeneous and particular solution equations, nonlinear equations, first-order, second-order, third-order, and many other equations.

### How to calculate a differential equation on dCode?

The equation must follow a strict syntax to get a solution in the differential equation solver:

— Use `′` (single quote) to represent the derivative of order 1, `′′` for the derivative of order 2, `′′′` for the derivative of order 3, etc.

__Example:__ `f' + f = 0`

— Do not indicate the variable to derive in the diffequation.

__Example:__ `f(x)` is noted `f` and the variable `x` must be specified in the variable input.

__Example:__ $ f' + f = 1 \Rightarrow f(x) = c_1 e^{-x}+1 $ with $ c_1 $ a constant

— Only the function is differentiable and not a combination of function

__Example:__ `(1/f)'` is invalid but `1/(f')` is correct

### How to add initial values/conditions?

It is possible to add one or more initial conditions in the corresponding box by adding the logical operator `&&` between 2 equations.

__Example:__ Write `f'(0)=-1 && f(1)=0`

### What is the difference between a general solution and a particular solution?

The general solution of a differential equation gives an overview of all possible solutions (by integrating `c` constants) presented in a general form that can encompass an infinite range of solutions.

The particular solution is a particular solution, obtained by setting the constants to particular values meeting the initial conditions defined by the user or by the context of the problem.

Sometimes dCode will not be able to calculate the general solution but will be able to find one or more particular solutions.

### How to find values of constants c?

Use known information about the function and its derivative(s) as the initial conditions of the system.

__Example:__ The position of an object is $ h $ at the start of an experiment, write something like $ f(0) = h $

__Example:__ Object speed is $ 0 $ after $ n $ seconds, write something like $ f'(n) = 0 $

### What are the notations of the differential equations?

There are multiple notations for a function `f`:

__Example:__ $$ f'(x) = \frac{\mathrm{d} f(x)}{\mathrm{d}x} $$

__Example:__ $$ f''(x) = \frac{\mathrm{d}^2 f(x)}{\mathrm{d}x^2} $$

The apostrophe indicates the order/degree of derivation, the letter in parenthesis is the derivation variable.

The exponent indicates the order/degree of derivation, the letter of the denominator is the derivation variable.

### How to solve a differential equation step by step?

The calculation steps of the dCode solver are not displayed because they are computer operations far from the steps of a student's process.

## Source code

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Cite as source (bibliography):

*Differential Equation Solver* on dCode.fr [online website], retrieved on 2024-07-31,