3.2E: The Improved Euler Method (Exercises)
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The following exercises involve initial value problems considered in the exercises in Section 3.1. You’ll find it instructive to compare the results that you obtain here with the corresponding results that you obtained in Section 3.1.
You will want to save the results of these exercises, along with those of section 3.1, since we will revisit them in the next section.
In Exercises 1-5 use the improved Euler 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. \(y'+x^2y=\sin xy,\quad y(1)=\pi;\quad h=0.2\)
6. Use the improved Euler method with step size \(h=0.1\) to find approximate values of the solution of the initial value problem \[y'+3y=7e^{4x},\quad y(0)=2\nonumber \] at \(x=0\), \(0.1\), \(0.2\), \(0.3\), …, \(1.0\). Compare these approximate values with the values of the exact solution \(y=e^{4x}+e^{-3x}\), which can be obtained by the method of Section 2.3. Present your results in a table like Table 3.2.2.
7. Use the improved Euler method with step size \(h=0.1\) to find approximate values of the solution of the initial value problem \[y'+{2\over x}y={3\over x^3}+1,\quad y(1)=1\nonumber \] at \(x=1.0\), \(1.1\), \(1.2\), \(1.3\), …, \(2.0\). Compare these approximate values with the values of the exact solution \[y={1\over3x^2}(9\ln x+x^3+2)\nonumber \] which can be obtained by the method of Section 2.3. Present your results in a table like Table 3.2.2.
8. Use the improved Euler method with step size \(h=0.05\) to find approximate values of the solution of the initial value problem \[y'={y^2+xy-x^2\over x^2},\quad y(1)=2,\nonumber \] at \(x=1.0\), \(1.05\), \(1.10\), \(1.15\), …, \(1.5\). Compare these approximate values with the values of the exact solution \[y={x(1+x^2/3)\over1-x^2/3}\nonumber \] obtained by a method in Section 2.4. Present your results in a table like Table 3.2.2.
9. It was shown in an example in Section 2.1 that \[y^5+y=x^2+x-4\nonumber \] is an implicit solution of the initial value problem \[y'={2x+1\over5y^4+1},\quad y(2)=1. \tag{A}\] Use the improved Euler method with step size \(h=0.1\) to find approximate values of the solution of (A) at \(x=2.0\), \(2.1\), \(2.2\), \(2.3\), …, \(3.0\). Present your results in tabular form. To check the error in these approximate values, construct another table of values of the residual \[R(x,y)=y^5+y-x^2-x+4\nonumber \] for each value of \((x,y)\) appearing in the first table.
10. It was previously shown that \[x^4y^3+x^2y^5+2xy=4\nonumber \] is an implicit solution of the initial value problem \[y'=-{4x^3y^3+2xy^5+2y\over3x^4y^2+5x^2y^4+2x},\quad y(1)=1. \tag{A} \] Use the improved Euler method with step size \(h=0.1\) to find approximate values of the solution of (A) at \(x=1.0\), \(1.14\), \(1.2\), \(1.3\), …, \(2.0\). Present your results in tabular form. To check the error in these approximate values, construct another table of values of the residual \[R(x,y)=x^4y^3+x^2y^5+2xy-4\nonumber \] for each value of \((x,y)\) appearing in the first table.
11. Use the improved Euler method with step size \(h=0.1\) to find approximate values of the solution of the initial value problem \[(3y^2+4y)y'+2x+\cos x=0, \quad y(0)=1 \nonumber \] at \(x=0\), \(0.1\), \(0.2\), \(0.3\), …, \(1.0\).
12. Use the improved Euler method with step size \(h=0.1\) to find approximate values of the solution of the initial value problem \[y'+{(y+1)(y-1)(y-2)\over x+1}=0, \quad y(1)=0 \nonumber \] at \(x=1.0\), \(1.1\), \(1.2\), \(1.3\), …, \(2.0\).
13. Use the improved Euler method with step size \(h=0.1\) to find approximate values of the solution of the initial value problem \[y'+3y=e^{-3x}(1-2x),\quad y(0)=2,\nonumber \] at \(x=0\), \(0.1\), \(0.2\), \(0.3\), …, \(1.0\). Compare these approximate values with the values of the exact solution \(y=e^{-3x}(2+x-x^2)\), which can be obtained by the method of Section 2.3.
The linear initial value problems in Exercises 14-22 can’t be solved exactly in terms of known elementary functions. In each exercise, use the improved Euler’s method with the indicated step size to find approximate values of the solution of the given initial value problem at 11 equally spaced points (including the endpoints) in the interval.
14. \(y'-2y= {1\over1+x^2},\quad y(2)=2;\quad\) \(h=0.1\) on \([2,3]\)
15. \(y'+2xy=x^2,\quad y(0)=3;\quad\) \(h=0.2\) on \([0,2]\)
16. \( {y'+{1\over x}y={\sin x\over x^2},\quad y(1)=2;}\quad\) \(h=0.2\) on \([1,3]\)
17. \( {y'+y={e^{-x}\tan x\over x},\quad y(1)=0};\quad\) \(h=0.05\) on \([1,1.5]\)
18. \( {y'+{2x\over 1+x^2}y={e^x\over (1+x^2)^2}, \quad y(0)=1};\quad\) \(h=0.2\) on \([0,2]\)
19. \(xy'+(x+1)y=e^{x^2},\quad y(1)=2;\quad\) \(h=0.05\) on \([1,1.5]\)
20. \(y'+3y=xy^2(y+1),\quad y(0)=1;\quad\) \(h=0.1\) on \([0,1]\)
21. \( {y'-4y={x\over y^2(y+1)},\quad y(0)=1;}\quad\) \(h=0.1\) on \([0,1]\)
22. \( {y'+2y={x^2\over1+y^2},\quad y(2)=1;}\quad\) \(h=0.1,\) on \([2,3]\)