Question

a)Program a calculator or computer to use Euler's method to
compute *y*(1), where *y*(*x*) is the solution
of the given initial-value problem. (Give all answers to four
decimal places.)

dy |

dx |

+ 3*x*^{2}* y* =
9

* y*(0) = 4

h =
1 |
y(1) = |

h =
0.1 |
y(1) = |

h =
0.01 |
y(1) = |

h =
0.001 |
y(1) = |

(b) Verify that

* y* = 3 +

is the exact solution of the differential equation.

* y* = 3 +

⇒ * y'*
=

LHS = * y'* +
3

+ 3*x*^{2}(3 +
*e*^{−x3}) =
−3*x*^{2}*e*^{−x3}
+

+
3*x*^{2}*e*^{−x3}
= 9*x*^{2} = RHS

* y*(0) =

+ *e*^{−0} = 3 + 1 = 4

(c) Find the errors in using Euler's method to compute
*y*(1) with the step sizes in part (a). (Give all answers to
four decimal places.)

h =
1 |
error = (exact value − approximate value) = |

h =
0.1 |
error = (exact value − approximate value) = |

h =
0.01 |
error = (exact value − approximate value) = |

h =
0.001 |
error = (exact value − approximate value) = |

What happens to the error when the step size is divided by 10?

When the step size is divided by 10, the error estimate is ---Select--- divided by 5 multiplied by 10 divided by 10 multiplied by 5 divided by 100 (approximately).

Answer #1

Note: when h = 0.1, some part of calculation is

Use Euler's method to approximate y(0.2), where
y(x) is the solution of the initial-value
problem
y'' − 4y' + 4y = 0, y(0) = −3, y'(0) =
1.
Use
h = 0.1.
Find the analytic solution of the problem, and compare the
actual value of y(0.2) with y2. (Round
your answers to four decimal places.)
y(0.2)
≈
(Euler approximation)
y(0.2)
=
-2.3869
(exact value)
I'm looking for the Euler approximation number, thanks.

Use Euler's method to approximate y(1.2), where
y(x) is the solution of the initial-value
problem
x2y'' − 2xy' + 2y = 0, y(1) =
9, y'(1) = 9,
where
x > 0.
Use
h = 0.1.
Find the analytic solution of the problem, and compare the
actual value of y(1.2) with y2. (Round
your answers to four decimal places.)
y(1.2)
≈
(Euler approximation)
y(1.2)
=
(exact value)

Use Euler's method with step size 0.1 to estimate y(0.5), where
y(x) is the solution of the initial-value problem
y'=3x+y^2, y(0)=−1
y(0.5)=

Use Euler's method to approximate y(0.7), where y(x) is the
solution of the initial-value problem y'' − (2x + 1)y = 1, y(0) =
3, y'(0) = 1. First use one step with h = 0.7. (Round your answer
to
two decimal places.) y(0.7) = ? Then repeat the calculations
using two steps with h = 0.35. (Round your answers to two decimal
places.) y(0.35) = ? y(0.7) =?

Given the initial value problem:
y'=6√(t+y), y(0)=1
Use Euler's method with step size h = 0.1 to estimate:
y(0.1) =
y(0.2) =

Apply Euler's method twice to approximate the solution of the
equation y'=y-x-1, y(0)=1 at x=0.5. Use h=0.1.
a.
y(0.5)=1.089
b.
y(0.5)=0.579
c.
y(0.5)=1.534
d.
y(0.5)=0.889

Use Euler's method with step size 0.5 to compute the approximate
y-values y1 ≈ y(0.5),
y2 ≈ y(1), y3 ≈
y(1.5), and y4 ≈ y(2) of the
solution of the initial-value problem
y′ = 1 + 2x − 2y,
y(0)=1.
y1 =
y2 =
y3 =
y4 =

Use Euler's method with step size 0.2 to estimate y(0.6) where
y(x) is the solution to the initial value problem y' = y+x^2, y(0)
= 3

Use Euler's method with step size 0.5 to compute the approximate
y-values y1, y2, y3 and y4 of the solution of the initial-value
problem y' = y − 3x, y(4) = 0.y1 = y2 = y3 = y4 =

Use Euler's method with step size 0.5 to compute the approximate
y-values y1, y2,
y3 and y4 of the solution
of the initial-value problem
y' = y − 4x,
y(4) = 0.
y1 =
y2 =
y3 =
y4 =

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