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solve/parametrized

parametrize the solutions of a scalar functions of two variables

 Calling Sequence solve(eqn, varlist)

Parameters

 eqn - single equation in two variables varlist - list of two names applied to a parameter (for example, $\left[x\left(t\right),y\left(t\right)\right]$) or two equations of the form $\left[x\left(t\right)=\mathrm{x0},y\left(t\right)=\mathrm{y0}\right]$ giving the parametrization point. The desired name of the parameter to use in the solution (for example, t) appears as an argument to the variable names. Note that $t=0$ corresponds to $\mathrm{x0},\mathrm{y0}$.

Description

 • The solve(eqn,varlist) command substitutes $y=\mathrm{y0}+t\left(x-\mathrm{x0}\right)$ into the given function and tries to solve the resulting equation (set equal to zero) for $x$ as a function of $t$. If successful, this gives a parametrized solution of the original equation, in that $f\left(x\left(t\right),\mathrm{y0}+t\left(x\left(t\right)-\mathrm{x0}\right)\right)=0$.
 Note: This command does not find solutions of the form $x=at+\mathrm{x0},y=bt+\mathrm{y0}$.  It can sometimes find parametrized solutions of nonpolynomial equations, and can find nonrational parametrizations of polynomial equations.  To find rational parametrizations of polynomial systems, use the command algcurves[parametrization].

Examples

 > $\mathrm{solve}\left({u}^{2}+{v}^{2}-1,\left[u\left(s\right)=-1,v\left(s\right)=0\right]\right)$
 $\left[\left[{u}{=}\frac{{{s}}^{{2}}{-}{1}}{{{s}}^{{2}}{+}{1}}{,}{v}{=}\frac{{2}{}{s}}{{{s}}^{{2}}{+}{1}}\right]\right]$ (1)
 > $\mathrm{tacnode}≔2{x}^{4}-3{x}^{2}y+{y}^{4}-2{y}^{3}+{y}^{2}$
 ${\mathrm{tacnode}}{≔}{2}{}{{x}}^{{4}}{+}{{y}}^{{4}}{-}{3}{}{{x}}^{{2}}{}{y}{-}{2}{}{{y}}^{{3}}{+}{{y}}^{{2}}$ (2)
 > $\mathrm{tacsol}≔\mathrm{solve}\left(\mathrm{tacnode},\left[x\left(t\right),y\left(t\right)\right]\right)$
 ${\mathrm{tacsol}}{≔}\left[\left[{x}{=}{-}\frac{\left({-}{2}{}{{t}}^{{2}}{+}\sqrt{{12}{}{{t}}^{{2}}{+}{1}}{-}{3}\right){}{t}}{{2}{}\left({{t}}^{{4}}{+}{2}\right)}{,}{y}{=}{-}\frac{{{t}}^{{2}}{}\left({-}{2}{}{{t}}^{{2}}{+}\sqrt{{12}{}{{t}}^{{2}}{+}{1}}{-}{3}\right)}{{2}{}\left({{t}}^{{4}}{+}{2}\right)}\right]{,}\left[{x}{=}\frac{\left({2}{}{{t}}^{{2}}{+}{3}{+}\sqrt{{12}{}{{t}}^{{2}}{+}{1}}\right){}{t}}{{2}{}\left({{t}}^{{4}}{+}{2}\right)}{,}{y}{=}\frac{{{t}}^{{2}}{}\left({2}{}{{t}}^{{2}}{+}{3}{+}\sqrt{{12}{}{{t}}^{{2}}{+}{1}}\right)}{{2}{}\left({{t}}^{{4}}{+}{2}\right)}\right]\right]$ (3)
 > $\mathrm{solve}\left({x}^{x}-{y}^{y},\left[x\left(t\right),y\left(t\right)\right]\right)$
 $\left[\left[{x}{=}{{ⅇ}}^{\frac{{t}{}{\mathrm{ln}}{}\left(\frac{{1}}{{t}}\right)}{{t}{-}{1}}}{,}{y}{=}{t}{}{{ⅇ}}^{\frac{{t}{}{\mathrm{ln}}{}\left(\frac{{1}}{{t}}\right)}{{t}{-}{1}}}\right]\right]$ (4)

References

 Corless, Robert M. Essential Maple 7. Chap. 3. Springer-Verlag.
 Hardy, G. H. Pure Mathematics. Cambridge University Press, 1952.