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Magazine Problem 1
In each of the following exercises, use Euler's method with the prescribed \(\Delta x\) to approximate the solution of the initial value problem in the given interval. In Exercises 1 through \(6,\) solve the problem by elementary methods and compare the approximate values of \(y\) with the correct values. $$ y^{\prime}=x+y ; \quad \text { when } x=0, y=1 ; \quad \Delta x=0.1 \text { and } 0 \leq x \leq 1 $$
Problem 3
In each of the following exercises, use Euler's method with the prescribed \(\Delta x\) to approximate the solution of the initial value problem in the given interval. In Exercises 1 through \(6,\) solve the problem by elementary methods and compare the approximate values of \(y\) with the correct values. $$ y^{\prime}=x+y ; \quad \text { when } x=0, y=2 ; \quad \Delta x=0.1 \text { and } 0 \leq x \leq 1 $$
Problem 4
In each of the following exercises, use Euler's method with the prescribed \(\Delta x\) to approximate the solution of the initial value problem in the given interval. In Exercises 1 through \(6,\) solve the problem by elementary methods and compare the approximate values of \(y\) with the correct values. $$ y^{\prime}=x+y ; \quad \text { when } x=1, y=1 ; \quad \Delta x=0.1 \text { and } 1 \leq x \leq 2 . $$
Problem 5
In each of the following exercises, use Euler's method with the prescribed \(\Delta x\) to approximate the solution of the initial value problem in the given interval. In Exercises 1 through \(6,\) solve the problem by elementary methods and compare the approximate values of \(y\) with the correct values. $$ y^{\prime}=x+y ; \quad \text { when } x=2, y=-1 ; \quad \Delta x=0.1 \text { and } 2 \leq x \leq 3 $$
Problem 6
In each of the following exercises, use Euler's method with the prescribed \(\Delta x\) to approximate the solution of the initial value problem in the given interval. In Exercises 1 through \(6,\) solve the problem by elementary methods and compare the approximate values of \(y\) with the correct values. $$ y^{\prime}=2 x-3 y ; \quad \text { when } x=0, y=2 ; \quad \Delta x=0.1 \text { and } 0 \leq x \leq 1 $$
Problem 6
\(y^{\prime}=y^{2}+x^{2} ; \quad\) when \(x=0, y=1 ; \quad \Delta x=0.1\) and \(0 \leq x \leq 0.5\)
Problem 7
\(y^{\prime}=y^{2}-x^{2} ; \quad\) when \(x=0, y=1 ; \quad \Delta x=0.1\) and \(0 \leq x \leq 0.5\)
Problem 7
In each of the following exercises, use Euler's method with the prescribed \(\Delta x\) to approximate the solution of the initial value problem in the given interval. In Exercises 1 through \(6,\) solve the problem by elementary methods and compare the approximate values of \(y\) with the correct values. $$ y^{\prime}=e^{-x y} ; \quad \text { when } x=0, y=0 ; \quad \Delta x=0.2 \text { and } 0 \leq x \leq 2 $$
Problem 8
Use Taylor's series to determine to three places the value of the solution of the problem $$y^{\prime}=-x y^{2} ; \quad \text { when } x=0, y=1$$ for \(x=0.1 .0 .2\). and \(0.3 .\) Compare your results with the values obtained by solving the problem by elementary means.
Problem 9
In each of the following exercises, use Euler's method with the prescribed \(\Delta x\) to approximate the solution of the initial value problem in the given interval. In Exercises 1 through \(6,\) solve the problem by elementary methods and compare the approximate values of \(y\) with the correct values. $$ y^{\prime}=\left(1+x^{2}+y^{2}\right)^{-1} ; \quad \text { when } x=0, y=0 ; \quad \Delta x=0.2 \text { and } 0 \leq x \leq 2 $$
