# Differential Equations MATLAB Homework Help

A **differential equation** is a mathematical equation that relates an unknown function of one or more variables that with its derivatives. The differential equations arise whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. Due to such relations are extremely common, the differential equations play a prominent role in many disciplines including engineering, the physics, the economics, the biology.

Differential equations are mathematically studied from several different perspectives, which mostly concerned with their solutions the set of functions that satisfy the equation. The simplest differential equations admit solutions given by explicit formulas; some properties of solutions of a given differential equation may be determined without finding their exact form. In case if a self-contained formula for the solution is not available, solution may be numerically approximated using computers. Theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, where many numerical methods have been developed to determine solutions with a given degree of accuracy.

**Function to solve the differential equation:**

**Dsolve**

## Syntax

`S = dsolve(eqn)`

`S = dsolve(eqn,cond)`

`S = dsolve(eqn,cond,Name,Value)`

`Y = dsolve(eqns)`

`Y = dsolve(eqns,conds)`

`Y = dsolve(eqns,conds,Name,Value)`

`[y1,...,yN] = dsolve(eqns)`

`[y1,...,yN] = dsolve(eqns,conds)`

`[y1,...,yN] = dsolve(eqns,conds,Name,Value)`

## Description

`S = dsolve(eqn)` solves the ordinary differential equation `eqn`. `eqn` is a symbolic equation containing `diff` to indicate derivatives. You can use a string with the letter `D` indicating derivatives. For example, `syms y(x); dsolve(diff(y) == y + 1)` and `dsolve('Dy = y + 1','x')` both solve the equation `dy/dx = y + 1` with respect to the variable `x`. `eqn` can be an array of such equations or strings.

`S = dsolve(eqn,cond)` solves the ordinary differential equation `eqn` with the initial or boundary condition `cond`.

`S = dsolve(eqn,cond,Name,Value)` uses additional options specified by one or more `Name,Value` pair arguments.

`Y = dsolve(eqns)` solves the system of ordinary differential equations `eqns` and returns a structure array that contains the solutions. Number of fields in the structure array corresponds to the number of independent variables in the system.

`Y = dsolve(eqns,conds)` solves the system of ordinary differential equations `eqns` with the initial or boundary conditions `conds`.

`Y = dsolve(eqns,conds,Name,Value)` which uses additional options specified by one or more `Name,Value` pair arguments.

`[y1,...,yN] = dsolve(eqns)` solves the system of ordinary differential equations `eqns` and assigns the solutions to the variables `y1,...,yN`.

`[y1,...,yN] = dsolve(eqns,conds)` solves the system of ordinary differential equations `eqns` with the initial or boundary conditions `conds`.

`[y1,...,yN] = dsolve(eqns,conds,Name,Value)` uses additional options specified by one or more `Name,Value` pair arguments.

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