CauchyEulerEquations - Maple Help

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ODE Steps for Cauchy-Euler Equations

Overview

 • This help page gives a few examples of using the command ODESteps to solve Cauchy-Euler equations.
 • See Student[ODEs][ODESteps] for a general description of the command ODESteps and its calling sequence.

Examples

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{ODEs}\right):$
 > $\mathrm{ode1}≔{x}^{2}\mathrm{diff}\left(y\left(x\right),x,x\right)-4x\mathrm{diff}\left(y\left(x\right),x\right)+2y\left(x\right)=0$
 ${\mathrm{ode1}}{≔}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{4}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{2}{}{y}{}\left({x}\right){=}{0}$ (1)
 > $\mathrm{ODESteps}\left(\mathrm{ode1}\right)$
 $\begin{array}{lll}{}& {}& \text{Let's solve}\\ {}& {}& {{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{4}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{2}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& {\text{Highest derivative means the order of the ODE is}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{2}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ \text{•}& {}& \text{Isolate 2nd derivative}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{{2}{}{y}{}\left({x}\right)}{{{x}}^{{2}}}{+}\frac{{4}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{x}}\\ \text{•}& {}& {\text{Group terms with}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{y}{}\left({x}\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}\frac{{4}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{x}}{+}\frac{{2}{}{y}{}\left({x}\right)}{{{x}}^{{2}}}{=}{0}\\ \text{•}& {}& \text{Multiply by denominators of the ODE}\\ {}& {}& {{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{4}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{2}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Make a change of variables}\\ {}& {}& {t}{=}{\mathrm{ln}}{}\left({x}\right)\\ \text{▫}& {}& \text{Substitute the change of variables back into the ODE}\\ {}& \text{◦}& {\text{Calculate the 1st derivative of}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{y}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{with respect to}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{x}{\text{, using the chain rule}}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ {}& \text{◦}& \text{Compute derivative}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}\frac{\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)}{{x}}\\ {}& \text{◦}& {\text{Calculate the 2nd derivative of}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{y}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{with respect to}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{x}{\text{, using the chain rule}}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ {}& \text{◦}& \text{Compute derivative}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}{-}\frac{\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)}{{{x}}^{{2}}}{+}\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)}{{{x}}^{{2}}}\\ {}& {}& \text{Substitute the change of variables back into the ODE}\\ {}& {}& {{x}}^{{2}}{}\left({-}\frac{\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)}{{{x}}^{{2}}}{+}\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)}{{{x}}^{{2}}}\right){-}{4}{}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){+}{2}{}{y}{}\left({t}\right){=}{0}\\ \text{•}& {}& \text{Simplify}\\ {}& {}& {-}{5}{}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){+}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){+}{2}{}{y}{}\left({t}\right){=}{0}\\ \text{•}& {}& \text{Characteristic polynomial of ODE}\\ {}& {}& {{r}}^{{2}}{-}{5}{}{r}{+}{2}{=}{0}\\ \text{•}& {}& {\text{Use quadratic formula to solve for}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{r}\\ {}& {}& {r}{=}\frac{{5}{±}\left(\left[{}\right]\right)}{{2}}\\ \text{•}& {}& \text{Roots of the characteristic polynomial}\\ {}& {}& {r}{=}\left(\frac{{5}}{{2}}{-}\frac{\sqrt{{17}}}{{2}}{,}\frac{{5}}{{2}}{+}\frac{\sqrt{{17}}}{{2}}\right)\\ \text{•}& {}& \text{1st solution of the ODE}\\ {}& {}& {{y}}_{{1}}{}\left({t}\right){=}{{ⅇ}}^{\left(\frac{{5}}{{2}}{-}\frac{\sqrt{{17}}}{{2}}\right){}{t}}\\ \text{•}& {}& \text{2nd solution of the ODE}\\ {}& {}& {{y}}_{{2}}{}\left({t}\right){=}{{ⅇ}}^{\left(\frac{{5}}{{2}}{+}\frac{\sqrt{{17}}}{{2}}\right){}{t}}\\ \text{•}& {}& \text{General solution of the ODE}\\ {}& {}& {y}{}\left({t}\right){=}{\mathrm{_C1}}{}{{y}}_{{1}}{}\left({t}\right){+}{\mathrm{_C2}}{}{{y}}_{{2}}{}\left({t}\right)\\ \text{•}& {}& \text{Substitute in solutions}\\ {}& {}& {y}{}\left({t}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{\left(\frac{{5}}{{2}}{-}\frac{\sqrt{{17}}}{{2}}\right){}{t}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{\left(\frac{{5}}{{2}}{+}\frac{\sqrt{{17}}}{{2}}\right){}{t}}\\ \text{•}& {}& {\text{Change variables back using}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{t}{=}{\mathrm{ln}}{}\left({x}\right)\\ {}& {}& {y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{\left(\frac{{5}}{{2}}{-}\frac{\sqrt{{17}}}{{2}}\right){}\left[{}\right]}{+}{\mathrm{_C2}}{}{{ⅇ}}^{\left(\frac{{5}}{{2}}{+}\frac{\sqrt{{17}}}{{2}}\right){}\left[{}\right]}\\ \text{•}& {}& \text{Simplify}\\ {}& {}& {y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}\frac{\left({-}{5}{+}\sqrt{{17}}\right){}\left[{}\right]}{{2}}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{\frac{\left({5}{+}\sqrt{{17}}\right){}\left[{}\right]}{{2}}}\end{array}$ (2)
 > $\mathrm{ode2}≔{x}^{3}\mathrm{diff}\left(y\left(x\right),x,x,x\right)+3{x}^{2}\mathrm{diff}\left(y\left(x\right),x,x\right)-6x\mathrm{diff}\left(y\left(x\right),x\right)-6y\left(x\right)=0$
 ${\mathrm{ode2}}{≔}{{x}}^{{3}}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{3}{}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{y}{}\left({x}\right){=}{0}$ (3)
 > $\mathrm{ODESteps}\left(\mathrm{ode2}\right)$
 $\begin{array}{lll}{}& {}& \text{Let's solve}\\ {}& {}& {{x}}^{{3}}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{3}{}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& {\text{Highest derivative means the order of the ODE is}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{3}\\ {}& {}& \frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ \text{•}& {}& \text{Isolate 3rd derivative}\\ {}& {}& \frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{6}{}{y}{}\left({x}\right)}{{{x}}^{{3}}}{-}\frac{{3}{}\left(\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{x}{-}{2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{{x}}^{{2}}}\\ \text{•}& {}& {\text{Group terms with}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{y}{}\left({x}\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}}\\ {}& {}& \frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}\frac{{6}{}{y}{}\left({x}\right)}{{{x}}^{{3}}}{+}\frac{{3}{}\left(\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{x}{-}{2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{{x}}^{{2}}}{=}{0}\\ \text{•}& {}& \text{Multiply by denominators of the ODE}\\ {}& {}& {{x}}^{{3}}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{y}{}\left({x}\right){+}{3}{}{x}{}\left(\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{x}{-}{2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){=}{0}\\ \text{•}& {}& \text{Make a change of variables}\\ {}& {}& {t}{=}{\mathrm{ln}}{}\left({x}\right)\\ \text{▫}& {}& \text{Substitute the change of variables back into the ODE}\\ {}& \text{◦}& {\text{Calculate the 1st derivative of}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{y}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{with respect to}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{x}{\text{, using the chain rule}}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ {}& \text{◦}& \text{Compute derivative}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}\frac{\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)}{{x}}\\ {}& \text{◦}& {\text{Calculate the 2nd derivative of}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{y}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{with respect to}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{x}{\text{, using the chain rule}}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ {}& \text{◦}& \text{Compute derivative}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}{-}\frac{\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)}{{{x}}^{{2}}}{+}\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)}{{{x}}^{{2}}}\\ {}& \text{◦}& {\text{Calculate the 3rd derivative of}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{y}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{with respect to}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{x}{\text{, using the chain rule}}\\ {}& {}& \frac{{{ⅆ}}^{{3}}}{{ⅆ}{{t}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ {}& \text{◦}& \text{Compute derivative}\\ {}& {}& \frac{{{ⅆ}}^{{3}}}{{ⅆ}{{t}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}\frac{{2}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}{{{x}}^{{3}}}{-}\frac{{3}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}{{{x}}^{{3}}}{+}\frac{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{t}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)}{{{x}}^{{3}}}\\ {}& {}& \text{Substitute the change of variables back into the ODE}\\ {}& {}& {{x}}^{{3}}{}\left(\frac{{2}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}{{{x}}^{{3}}}{-}\frac{{3}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}{{{x}}^{{3}}}{+}\frac{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{t}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)}{{{x}}^{{3}}}\right){-}{6}{}{y}{}\left({t}\right){+}{3}{}{x}{}\left(\left({-}\frac{\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)}{{{x}}^{{2}}}{+}\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)}{{{x}}^{{2}}}\right){}{x}{-}\frac{{2}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}{{x}}\right){=}{0}\\ \text{•}& {}& \text{Simplify}\\ {}& {}& {-}{7}{}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){+}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{t}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){-}{6}{}{y}{}\left({t}\right){=}{0}\\ \text{▫}& {}& \text{Convert linear ODE into a system of first order ODEs}\\ {}& \text{◦}& {\text{Define new variable}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{{y}}_{{1}}{}\left({t}\right)\\ {}& {}& {{y}}_{{1}}{}\left({t}\right){=}{y}{}\left({t}\right)\\ {}& \text{◦}& {\text{Define new variable}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{{y}}_{{2}}{}\left({t}\right)\\ {}& {}& {{y}}_{{2}}{}\left({t}\right){=}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\\ {}& \text{◦}& {\text{Define new variable}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{{y}}_{{3}}{}\left({t}\right)\\ {}& {}& {{y}}_{{3}}{}\left({t}\right){=}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\\ {}& \text{◦}& {\text{Isolate for}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{3}}{}\left({t}\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{using original ODE}}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{3}}{}\left({t}\right){=}{7}{}{{y}}_{{2}}{}\left({t}\right){+}{6}{}{{y}}_{{1}}{}\left({t}\right)\\ {}& {}& \text{Convert linear ODE into a system of first order ODEs}\\ {}& {}& \left[{{y}}_{{2}}{}\left({t}\right){=}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{1}}{}\left({t}\right){,}{{y}}_{{3}}{}\left({t}\right){=}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{2}}{}\left({t}\right){,}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{3}}{}\left({t}\right){=}{7}{}{{y}}_{{2}}{}\left({t}\right){+}{6}{}{{y}}_{{1}}{}\left({t}\right)\right]\\ \text{•}& {}& \text{Define vector}\\ {}& {}& \stackrel{{\to }}{{y}}{}\left({t}\right){=}\left[\begin{array}{c}{{y}}_{{3}}{}\left({t}\right)\\ {{y}}_{{1}}{}\left({t}\right)\\ {{y}}_{{2}}{}\left({t}\right)\end{array}\right]\\ \text{•}& {}& \text{System to solve}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\stackrel{{\to }}{{y}}{}\left({t}\right){=}{A}{·}\stackrel{{\to }}{{y}}{}\left({t}\right)\\ \text{•}& {}& {\text{To solve the system find eigenvalues and eigenvectors of}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{A}\\ {}& {}& {A}{=}\left[\begin{array}{ccc}{0}& {6}& {7}\\ {0}& {0}& {1}\\ {1}& {0}& {0}\end{array}\right]\\ \text{•}& {}& \text{Eigenpairs of A}\\ {}& {}& \left[\left[{-1}{,}\left[\begin{array}{c}{-1}\\ {-1}\\ {1}\end{array}\right]\right]{,}\left[{3}{,}\left[\begin{array}{c}{3}\\ \frac{{1}}{{3}}\\ {1}\end{array}\right]\right]{,}\left[{-2}{,}\left[\begin{array}{c}{-2}\\ {-}\frac{{1}}{{2}}\\ {1}\end{array}\right]\right]\right]\\ \text{•}& {}& \text{Consider eigenpair}\\ {}& {}& \left[{-1}{,}\left[\begin{array}{c}{-1}\\ {-1}\\ {1}\end{array}\right]\right]\\ \text{•}& {}& \text{Solution to homogeneous system from eigenpair}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{1}}{}\left({t}\right){=}\left[{}\right]\\ \text{•}& {}& \text{Consider eigenpair}\\ {}& {}& \left[{3}{,}\left[\begin{array}{c}{3}\\ \frac{{1}}{{3}}\\ {1}\end{array}\right]\right]\\ \text{•}& {}& \text{Solution to homogeneous system from eigenpair}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{2}}{}\left({t}\right){=}\left[{}\right]\\ \text{•}& {}& \text{Consider eigenpair}\\ {}& {}& \left[{-2}{,}\left[\begin{array}{c}{-2}\\ {-}\frac{{1}}{{2}}\\ {1}\end{array}\right]\right]\\ \text{•}& {}& \text{Solution to homogeneous system from eigenpair}\\ {}& {}& {\stackrel{{\to }}{{y}}}_{{3}}{}\left({t}\right){=}\left[{}\right]\\ \text{•}& {}& \text{General solution to the system of ODEs}\\ {}& {}& \stackrel{{\to }}{{y}}{}\left({t}\right){=}{\mathrm{_C1}}{}{\stackrel{{\to }}{{y}}}_{{1}}{}\left({t}\right){+}{\mathrm{_C2}}{}{\stackrel{{\to }}{{y}}}_{{2}}{}\left({t}\right){+}{\mathrm{_C3}}{}{\stackrel{{\to }}{{y}}}_{{3}}{}\left({t}\right)\\ \text{•}& {}& \text{Substitute solutions into the general solution}\\ {}& {}& \stackrel{{\to }}{{y}}{}\left({t}\right){=}\left[{}\right]{+}\left[{}\right]{+}\left[{}\right]\\ \text{•}& {}& \text{First component of the vector is the solution to the ODE}\\ {}& {}& {y}{}\left({t}\right){=}{-}\left({-}{3}{}{\mathrm{_C2}}{}{{ⅇ}}^{{5}{}{t}}{+}{\mathrm{_C1}}{}{{ⅇ}}^{{t}}{+}{2}{}{\mathrm{_C3}}\right){}{{ⅇ}}^{{-}{2}{}{t}}\\ \text{•}& {}& {\text{Change variables back using}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{t}{=}{\mathrm{ln}}{}\left({x}\right)\\ {}& {}& {y}{}\left({x}\right){=}{-}\left({-}{3}{}{\mathrm{_C2}}{}{{ⅇ}}^{{5}{}\left[{}\right]}{+}{\mathrm{_C1}}{}{{ⅇ}}^{\left[{}\right]}{+}{2}{}{\mathrm{_C3}}\right){}{{ⅇ}}^{{-}{2}{}\left[{}\right]}\\ \text{•}& {}& \text{Simplify}\\ {}& {}& {y}{}\left({x}\right){=}{3}{}{\mathrm{_C2}}{}{{x}}^{{3}}{-}\frac{{\mathrm{_C1}}}{{x}}{-}\frac{{2}{}{\mathrm{_C3}}}{{{x}}^{{2}}}\end{array}$ (4)

 See Also