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# Euler Method Calculator

Enter the function, put the required points, hit calculate button to find approximated values using Euler method calculator

## Euler Method Calculator

Euler Method Calculator is a tool that is used to evaluate the solution of different functions or equations using the **Euler method**.

## What is meant by an Euler method?

The **Euler Method** is a numerical technique used to approximate the solutions of different equations. In the **18 ^{th} century** Swiss mathematician

**Euler**introduced this method due to this given the named Euler Method. The Euler Method is particularly useful when there is no analytical solution available for a given equation.

It is a basic numerical method and may not always provide the most accurate results, particularly for complex problems or greater functions with dependence on initial conditions.

**Initial Condition:**

Put the value of the **initial point **to start the iterative method and change the value of the point to continue the iteration at the selecting steps. The initial point value is shown as the (**t _{0}, y_{0}**) where “

**y**” is the value of the dependent variable of “

_{0}**y**” at a given initial point “

**t**”.

_{0}**Step Size:**

For the easiness of a difficult problem choose a small step size (**h**) that determines the intervals at which approximate the value of the solution. The smaller the step size increases the accuracy rate and also increases the computation time.

## Formula:

The mathematical formula of the iterative Euler method is stated below.

**y _{n+1} = y_{n}+ h × f(t_{n},y_{n})**

Where,

**y**= next iterative number_{n+1}**y**= initial iteration number_{n}**h**= step size of the interval**f(t**= Value of the function putting its initial values_{n},y_{n})

## How to solve problems with the Euler method?

**Example 1:**

Find the value of the given function if “**f(t,y) = t + 4y −3” **using the Euler method while the data are given below.

**h=4.0, t _{0}=3.0, y_{0}=5.0**.

**Solution:**

**Step 1: **Write the data.

h=4.0, t_{0}=3.0, y_{0}=5.0

f(t,y)= t + 4y −3

**Step 2: **Write the general formula of the Euler method.

`y`

_{n+1} = y_{n}+ h × f(t_{n},y_{n})

**Step 3: **Put the all above values in the above formula and simplify.

h=4.0, t_{0}=3.0, y_{0}=5.0

First start the iteration and put the “**n=0**” in the general formula it becomes.

`y`

_{0+1} = y_{0} + h × f(t_{0},y_{0})

y_{1} = y_{0} + h × f(t_{0},y_{0})

y_{1}=5.0 + 4.0 × 20.0

**y _{1}= 85.0**

**Example 2:**

Find the value of the given function if “**f(t,y)= t + 4y” **using the Euler method while the data are given below.

**h=2.0, t _{0}=3.0, y_{0}=4.0**.

**Solution:**

**Step 1: **Write the data.

h=2.0, t_{0}=3.0, y_{0}=4.0

f(t,y)= t + 4y

**Step 2: **Write the general formula of the Euler method.

`y`

_{n+1} = y_{n}+ h × f(t_{n},y_{n})

**Step 3: **Put the all above values in the above formula and simplify.

h=2.0, t_{0}=3.0, y_{0}=4.0

First start the iteration and put the “**n=0**” in the general formula it becomes.

`y`

_{0+1} = y_{0} + h × f(t_{0},y_{0})

y_{1} = y_{0} + h × f(t_{0},y_{0})

y_{1}= 4.0 + 2.0 × 19.0

y_{1}= 4.0 + 2.0 × 19.0

**y _{1}= 42.0**

**Step 4: **To find the next iteration find the values of t_{1}.

`t`

_{1 }= t_{0} + h

t_{1 }= 3.0 + 2.0

**t _{1} = 5.0**

`f(t`

_{1},y_{1}) = t_{1} + 4y_{1}

f(5,42) = 5.0 + (4)(42.0)

f(5,42) = 5.0 + 168.0

**f(5,42) = 173.0**

For the second iteration put the “**n=1**” in the general formula it becomes and put the all above values.

t_{1} = 5.0, y_{1}= 42.0, f(t_{1},y_{1})=173.0

`y`

_{1+1} = y_{1} + h × f(t_{1},y_{1})

y_{2} = y_{1} + h × f(t_{1},y_{1})

y_{2 }= 42.0 + 2.0×173.0

**y _{2 }= 388.0**