5.38 How to solve a set of equations for the derivative?

Written Feb 20, 2004

This is problem 7.4 chapter 4, in the Mary Boas book. Given

\begin{align*} x s^2+y t^2 &= 1\\ x^2 s+y^2 t &= xy-4 \end{align*}

Find \(\frac {dx}{dt}, \frac {dx}{ds}, \frac {dy}{dt}, \frac {dy}{ds}\) at \(x=1,y=-3,s=2,t=-1\)

This is how I did it in maple:

restart; 
alias(x=x(s,t)); 
alias(y=y(s,t)); 
alias(Xt= diff(x(s,t), t)); 
alias(Xs= diff(x(s,t), s)); 
alias(Yt= diff(y(s,t), t)); 
alias(Ys= diff(y(s,t), s)); 
 
eq1:= x*s^2+y*t^2=1; 
eq2:= x^2*s+y^2*t=x*y-4; 
 
r1:=diff(eq1,t); 
r2:=diff(eq1,s); 
r3:=diff(eq2,t); 
r4:=diff(eq2,s); 
 
sol:=solve({r1,r2,r3,r4},{Xt,Xs,Yt,Ys});
 

\begin{align*} {\frac {\partial }{\partial s}}x \left ( s,t \right ) &= -{\frac {x \left ( s,t \right ) \left ( x \left ( s,t \right ) {t}^{2}-4\,y \left ( s,t \right ) st+2\,x \left ( s,t \right ) s \right ) }{2\,x \left ( s,t \right ) s{t}^{2}-2\,y \left ( s,t \right ) t{s}^{2}+x \left ( s,t \right ) {s}^{2}-y \left ( s,t \right ) {t}^{2}}}\\ {\frac {\partial }{\partial t}}x \left ( s,t \right ) &=-{\frac {y \left ( s,t \right ) t \left ( -3\,y \left ( s,t \right ) t+2\,x \left ( s,t \right ) \right ) }{2\,x \left ( s,t \right ) s{t}^{2}-2\,y \left ( s,t \right ) t{ s}^{2}+x \left ( s,t \right ) {s}^{2}-y \left ( s,t \right ) {t}^{2}}}\\ {\frac {\partial }{\partial s}}y \left ( s,t \right ) &=-{\frac {x \left ( s,t \right ) \left ( 3\,x \left ( s,t \right ) s-2\,y \left ( s,t \right ) \right ) s}{2\,x \left ( s,t \right ) s{t}^{2}-2\,y \left ( s,t \right ) t {s}^{2}+x \left ( s,t \right ) {s}^{2}-y \left ( s,t \right ) {t}^{2}}}\\ {\frac {\partial }{\partial t}}y \left ( s,t \right ) &=-{\frac {y \left ( s,t \right ) \left ( 4\,x \left ( s,t \right ) st-y \left ( s,t \right ) {s }^{2}-2\,y \left ( s,t \right ) t \right ) }{2\,x \left ( s,t \right ) s{t} ^{2}-2\,y \left ( s,t \right ) t{s}^{2}+x \left ( s,t \right ) {s}^{2}-y \left ( s,t \right ) {t}^{2}}} \end{align*}

points:= {x=1,y=-3,s=2,t=-1}; 
subs(points,sol);