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].

$\mathrm{solve}\left(u^2+v^2-1\,\left[u\left(s\right)=-1\,v\left(s\right)=0\right]\right)$

$\left[\left[u=\frac{s^2-1}{s^2+1}\,v=\frac{2s}{s^2+1}\right]\right]$

$\mathrm{tacnode}≔2x^4-3x^{2}y+y^4-2y^3+y^2$

$\mathrm{tacnode}≔2x^4+y^4-3x^{2}y-2y^3+y^2$

$\mathrm{tacsol}≔\mathrm{solve}\left(\mathrm{tacnode}\,\left[x\left(t\right)\,y\left(t\right)\right]\right)$

$\mathrm{tacsol}≔\left[\left[x=-\frac{\left(-2t^2+\sqrt{12t^2+1}-3\right)t}{2\left(t^4+2\right)}\,y=-\frac{t^2\left(-2t^2+\sqrt{12t^2+1}-3\right)}{2\left(t^4+2\right)}\right]\,\left[x=\frac{\left(2t^2+3+\sqrt{12t^2+1}\right)t}{2\left(t^4+2\right)}\,y=\frac{t^2\left(2t^2+3+\sqrt{12t^2+1}\right)}{2\left(t^4+2\right)}\right]\right]$

$\mathrm{solve}\left(x^x-y^y\,\left[x\left(t\right)\,y\left(t\right)\right]\right)$

$\left[\left[x={\ⅇ}^{\frac{\ln \left(\frac{1}{t}\right)t}{t-1}}\,y=t{\ⅇ}^{\frac{\ln \left(\frac{1}{t}\right)t}{t-1}}\right]\right]$

Corless, Robert M. Essential Maple 7. Chap. 3. Springer-Verlag.

Hardy, G. H. Pure Mathematics. Cambridge University Press, 1952.