2.7E: Numerical Methods- Euler’s Method (Exercises)

In Exercises 1-5,  use Euler’s method to find approximate values of the solution of the given initial value problem at the points \(x_i=x_0+ih\), where \(x_0\) is the point where the initial condition is imposed and \(i=1\), \(2\), \(3\).

1. \(y'=2x^2+3y^2-2,\quad y(2)=1;\quad h=0.05\)

2. \(y'=y+\sqrt{x^2+y^2},\quad y(0)=1;\quad h=0.1\)

3. \(y'+3y=x^2-3xy+y^2,\quad y(0)=2;\quad h=0.05\)

4. \(y'= {1+x\over1-y^2},\quad y(2)=3;\quad h=0.1\)

5. \(y'+x^2y=\sin xy,\quad y(1)=\pi;\quad h=0.2\)

In Exercises 6, use GeoGebra numerical solver for Euler's method with step sizes \(h=0.1\), \(h=0.05\), and \(h=0.025\) to find approximate values of the solution of the initial value problem

6. \(y'+3y=7e^{4x},\quad y(0)=2 \quad\) at \(x=0\), \(0.1\), \(0.2\), \(0.3\), …, \(1.0\).

   Compare these approximate values with the values of the exact solution \(y=e^{4x}+e^{-3x}\)

In Exercises 7, use GeoGebra numerical solver for Euler's method with step sizes \(h=0.1\), \(h=0.05\), and \(h=0.025\) to find approximate values of the solution of the initial value problem

7. \(y'+{2\over x}y={3\over x^3}+1,\quad y(1)=1 \quad\) at \(x=1.0\), \(1.1\), \(1.2\), \(1.3\), …, \(2.0\).

   Compare these approximate values with the values of the exact solution \(y={1\over 3x^2}(9\ln x+x^3+2)\)

In Exercises 8, use GeoGebra numerical solver for Euler's method with step sizes \(h=0.05\), \(h=0.025\), and \(h=0.0125\) to find approximate values of the solution of the initial value problem

8. \(y'={y^2+xy-x^2\over x^2},\quad y(1)=2 \quad\) at \(x=1.0\), \(1.05\), \(1.10\), \(1.15\), …, \(1.5\).

   Compare these approximate values with the values of the exact solution \(y={x(1+x^2/3)\over1-x^2/3}\).

 

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