# Find $y{\left(\frac{1}{2} \right)}$ for $y^{\prime } = e^{- t^{2}}$, when $y{\left(0 \right)} = 1$, $h = \frac{1}{10}$ using the modified Euler's method

The calculator will find $y{\left(\frac{1}{2} \right)}$ for $y^{\prime } = e^{- t^{2}}$, when $y{\left(0 \right)} = 1$, $h = \frac{1}{10}$ using the modified Euler's method, with steps shown.

Related calculators: Euler's Method Calculator, Improved Euler (Heun's) Method Calculator

Or $y^{\prime } = f{\left(x,y \right)}$.
Or $x_{0}$.
$y_0=y(t_0)$ or $y_0=y(x_0)$.
Or $x_{1}$.

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Find $y{\left(\frac{1}{2} \right)}$ for $y^{\prime } = e^{- t^{2}}$, when $y{\left(0 \right)} = 1$, $h = \frac{1}{10}$ using the modified Euler's method.

### Solution

The modified Euler's method states that $y_{n+1} = y_{n} + h f{\left(t_{n} + \frac{h}{2},y_{n} + \frac{h}{2} f{\left(t_{n},y_{n} \right)} \right)}$, where $t_{n+1} = t_{n} + h$.

We have that $h = \frac{1}{10}$, $t_{0} = 0$, $y_{0} = 1$, and $f{\left(t,y \right)} = e^{- t^{2}}$.

### Step 1

$t_{1} = t_{0} + h = 0 + \frac{1}{10} = \frac{1}{10}$

$f{\left(t_{0},y_{0} \right)} = f{\left(0,1 \right)} = 1$

$y_{1} = y{\left(t_{1} \right)} = y{\left(\frac{1}{10} \right)} = y_{0} + h f{\left(t_{0} + \frac{h}{2},y_{0} + \frac{h}{2} f{\left(t_{0},y_{0} \right)} \right)} = 1 + \frac{f{\left(0 + \frac{\frac{1}{10}}{2},1 + \frac{\frac{1}{10}}{2} \cdot 1 \right)}}{10} = 1.09975031223975$

### Step 2

$t_{2} = t_{1} + h = \frac{1}{10} + \frac{1}{10} = \frac{1}{5}$

$f{\left(t_{1},y_{1} \right)} = f{\left(\frac{1}{10},1.09975031223975 \right)} = 0.990049833749168$

$y_{2} = y{\left(t_{2} \right)} = y{\left(\frac{1}{5} \right)} = y_{1} + h f{\left(t_{1} + \frac{h}{2},y_{1} + \frac{h}{2} f{\left(t_{1},y_{1} \right)} \right)} = 1.09975031223975 + \frac{f{\left(\frac{1}{10} + \frac{\frac{1}{10}}{2},1.09975031223975 + \frac{\frac{1}{10}}{2} \cdot 0.990049833749168 \right)}}{10} = 1.19752543595908$

### Step 3

$t_{3} = t_{2} + h = \frac{1}{5} + \frac{1}{10} = \frac{3}{10}$

$f{\left(t_{2},y_{2} \right)} = f{\left(\frac{1}{5},1.19752543595908 \right)} = 0.960789439152323$

$y_{3} = y{\left(t_{3} \right)} = y{\left(\frac{3}{10} \right)} = y_{2} + h f{\left(t_{2} + \frac{h}{2},y_{2} + \frac{h}{2} f{\left(t_{2},y_{2} \right)} \right)} = 1.19752543595908 + \frac{f{\left(\frac{1}{5} + \frac{\frac{1}{10}}{2},1.19752543595908 + \frac{\frac{1}{10}}{2} \cdot 0.960789439152323 \right)}}{10} = 1.29146674224043$

### Step 4

$t_{4} = t_{3} + h = \frac{3}{10} + \frac{1}{10} = \frac{2}{5}$

$f{\left(t_{3},y_{3} \right)} = f{\left(\frac{3}{10},1.29146674224043 \right)} = 0.913931185271228$

$y_{4} = y{\left(t_{4} \right)} = y{\left(\frac{2}{5} \right)} = y_{3} + h f{\left(t_{3} + \frac{h}{2},y_{3} + \frac{h}{2} f{\left(t_{3},y_{3} \right)} \right)} = 1.29146674224043 + \frac{f{\left(\frac{3}{10} + \frac{\frac{1}{10}}{2},1.29146674224043 + \frac{\frac{1}{10}}{2} \cdot 0.913931185271228 \right)}}{10} = 1.37993733273478$

### Step 5

$t_{5} = t_{4} + h = \frac{2}{5} + \frac{1}{10} = \frac{1}{2}$

$f{\left(t_{4},y_{4} \right)} = f{\left(\frac{2}{5},1.37993733273478 \right)} = 0.852143788966211$

$y_{5} = y{\left(t_{5} \right)} = y{\left(\frac{1}{2} \right)} = y_{4} + h f{\left(t_{4} + \frac{h}{2},y_{4} + \frac{h}{2} f{\left(t_{4},y_{4} \right)} \right)} = 1.37993733273478 + \frac{f{\left(\frac{2}{5} + \frac{\frac{1}{10}}{2},1.37993733273478 + \frac{\frac{1}{10}}{2} \cdot 0.852143788966211 \right)}}{10} = 1.46160598099459$

$y{\left(\frac{1}{2} \right)}\approx 1.46160598099459$A