The dsolve command with the options numeric and method=gear or method=gear[choice] finds a numerical solution using a Gear single-step extrapolation method.

The choice for the gear method are bstoer and polyextr.  The default bstoer choice is a Burlirsch-Stoer rational extrapolation method and the polyextr choice uses polynomial extrapolation.

By setting infolevel[`dsolve/gear`] or infolevel[dsolve] to 2, information about the last mesh point evaluation is provided in the event of an error.

The following options are available for the gear method.

'output','known'

The 'output' option specifies the desired output from dsolve, and the known option specifies user-defined known functions. These options are discussed in dsolve[numeric].

'maxfun'

Specifies a maximum on the number of evaluations of the right-hand side of the first order ODE system (See dsolve[maxfun]). This option can be disabled by specifying $\mathrm{maxfun}=0$. By default this option is disabled.

'number', 'procedure', 'start', 'initial', and 'procvars'

These options are used to specify the IVP using procedures. For more information, see dsolve[numeric,IVP].

'startinit','implicit', and 'optimize'

These options control the method and behavior of the computation. For more information on the first two, see dsolve[numeric,IVP], for the last, see dsolve[numeric].

'abserr' and 'relerr'

Specify the desired accuracy of the solution, and are discussed in dsolve[Error_Control]. The default values for gear are $\mathrm{abserr}=1.\times {10}^{\mathrm{-7}}$ and $\mathrm{relerr}=1.\times {10}^{\mathrm{-7}}$.

'minstep', 'maxstep', and 'initstep'

Provide finer control over the step size used in the method, and are also discussed in dsolve[Error_Control]. By default $\mathrm{minstep}=0.00001$, $\mathrm{maxstep}$ is disabled, and $\mathrm{initstep}=0.00001$. When setting $\mathrm{minstep}$ ensure that the value is adequately small to allow for the first few steps.

'maxord'

A positive integer, specifying the maximum order of extrapolation allowed. It must be less than $11$ (values of $6$ or less are recommended). The default value is $4$.

'maxpts'

A positive integer, specifying the maximum number of different single unit step sizes used in the extrapolation process (values less than $10$ are recommended). The default is $6$.

The computation may return with an error message corresponding to an error condition of the gear procedure.

Results can be plotted using the function odeplot in the plots package.

$\mathrm{Digits}≔10\:$

$\mathrm{deq1}≔\left\{\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,3\right)\right)=y\left(x\right)\mathrm{diff}\left(y\left(x\right)\,x\right)-x\right\}$

$\mathrm{deq1}≔\left\{\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)=y\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-x\right\}$

$\mathrm{init1}≔\left\{y\left(1\right)=2.4\,\mathrm{D}\left(y\right)\left(1\right)=3\,{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(1\right)=4\right\}$

$\mathrm{init1}≔\left\{y\left(1\right)=2.4\,\mathrm{D}\left(y\right)\left(1\right)=3\,{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(1\right)=4\right\}$

$\mathrm{ans1}≔\mathrm{dsolve}\left(\mathrm{deq1}\mathbf{union}\mathrm{init1}\,\mathrm{numeric}\,\mathrm{method}=\mathrm{gear}\left[\mathrm{polyextr}\right]\,\mathrm{initstep}=0.015\,\mathrm{minstep}=\mathrm{Float}\left(1\,-11\right)\,\mathrm{abserr}=\mathrm{Float}\left(1\,-5\right)\,\mathrm{relerr}=\mathrm{Float}\left(1\,-5\right)\right)\:$

$\mathrm{ans1}\left(1.01\right)$

$\left[x=1.01\,y\left(x\right)=2.43020104070930\,\frac{\ⅆ}{\ⅆx}y\left(x\right)=3.04031295467142\,\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=4.06288854913227\right]$

$\mathrm{dsys2}≔\left\{xy\left(x\right)-\sin \left(y\left(x\right)\right)=\mathrm{diff}\left(y\left(x\right)\,x\right)\,y\left(0\right)=-5.\right\}$

$\mathrm{dsys2}≔\left\{xy\left(x\right)-\sin \left(y\left(x\right)\right)=\frac{\ⅆ}{\ⅆx}y\left(x\right)\,y\left(0\right)=\mathrm{-5.}\right\}$

$\mathrm{ans2}≔\mathrm{dsolve}\left(\mathrm{dsys2}\,\mathrm{numeric}\,\mathrm{method}=\mathrm{gear}\,\mathrm{maxord}=3\right)\:$

$\mathrm{ans2}\left(0.45\right)$

$\left[x=0.45\,y\left(x\right)=\mathrm{-5.90260241046659}\right]$

$\mathrm{ans2}\left(0.9\right)$

$\left[x=0.9\,y\left(x\right)=\mathrm{-7.81336992184499}\right]$

$\mathrm{Digits}≔25\:$

$\mathrm{deq3}≔\mathrm{diff}\left(y\left(z\right)\,\mathrm{`\$`}\left(z\,3\right)\right)-y\left(z\right)z\mathrm{diff}\left(y\left(z\right)\,z\right)=5$

$\mathrm{deq3}≔\frac{{\ⅆ}^3}{\ⅆz^3}y\left(z\right)-y\left(z\right)z\left(\frac{\ⅆ}{\ⅆz}y\left(z\right)\right)=5$

$\mathrm{init3}≔y\left(0\right)=1,{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(0\right)=-1,\mathrm{D}\left(y\right)\left(0\right)=2$

$\mathrm{init3}≔y\left(0\right)=1,{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(0\right)=\mathrm{-1},\mathrm{D}\left(y\right)\left(0\right)=2$

$\mathrm{ans3}≔\mathrm{dsolve}\left(\left\{\mathrm{deq3}\,\mathrm{init3}\right\}\,\mathrm{numeric}\,\mathrm{method}=\mathrm{gear}\left[\mathrm{bstoer}\right]\,\mathrm{maxpts}=4\right)\:$

$\mathrm{ans3}\left(-1\right)$

$\left[z=\mathrm{-1.}\,y\left(z\right)=\mathrm{-2.348282662951510120622879}\,\frac{\ⅆ}{\ⅆz}y\left(z\right)=5.767096382629485105229235\,\frac{{\ⅆ}^2}{\ⅆz^2}y\left(z\right)=\mathrm{-8.239613529231847035339209}\right]$

$\mathrm{ans3}\left(0\right)$

$\left[z=0.\,y\left(z\right)=1.\,\frac{\ⅆ}{\ⅆz}y\left(z\right)=2.\,\frac{{\ⅆ}^2}{\ⅆz^2}y\left(z\right)=\mathrm{-1.}\right]$

Gear, C.W. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, 1971.