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factorout
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(%i1)globalsolve: true; (%o1) true (%i2) realonly: true; (%o2) true (%i3) /* Standard (x,y)-> (r,theta) coordinate translation */x(t) := r(t) * cos(theta(t)); (%o3) x(t) := r(t) cos(theta(t)) (%i4) y(t) := r(t) * sin(theta(t)); (%o4) y(t) := r(t) sin(theta(t)) (%i5) /* Constant course assumption */declare (slope,constant); (%o5) done (%i6) declare (slope,real); (%o6) done (%i7) declare (yintercept,constant); (%o7) done (%i8) declare (yintercept, real); (%o8) done (%i9) y(t)=slope * x(t) + yintercept; (%o9) r(t) sin(theta(t)) = slope r(t) cos(theta(t)) + yintercept (%i10) eq1: y(t)=slope * x(t) + yintercept; (%o10) r(t) sin(theta(t)) = slope r(t) cos(theta(t)) + yintercept (%i11) /* Constant speed assumption */declare (speedx,constant); (%o11) done (%i12) declare (speedx,real); (%o12) done (%i13) declare (speedy,constant); (%o13) done (%i14) declare (speedy,real); (%o14) done (%i15) declare (c1, constant); (%o15) done (%i16) declare (c1, real); (%o16) done (%i17) eq2: x(t) = speedx * (t + c1); (%o17) r(t) cos(theta(t)) = speedx (t + c1) (%i18) declare (c2, constant); (%o18) done (%i19) declare (c2, real); (%o19) done (%i20) eq3: y(t) = speedy * (t + c1) + c2*speedx; (%o20) r(t) sin(theta(t)) = speedy (t + c1) + c2 speedx (%i21) eq4: (rhs(eq3/eq2)); speedy (t + c1) + c2 speedx (%o21) --------------------------- speedx (t + c1) (%i22) eq5: factorout(eq4, speedx*(t+c1)); speedy (t + c1) + c2 speedx (%o22) --------------------------- speedx (t + c1) (%i23)
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expr: -((1+s*s*l*c)*((u-s*r)*(u+s*(w+p))-u*(u+s*w)))/((u+s*(w+p))*(s*l+(1+s*s*l*c)*s*r)); 2 (c l s + 1) ((u - r s) (s (w + p) + u) - u (s w + u)) (%o1) - ------------------------------------------------------ 2 (r s (c l s + 1) + l s) (s (w + p) + u) (%i2) factorout(expr,s); 2 (c l s + 1) ((u - r s) (s (w + p) + u) - u (s w + u)) (%o2) - ------------------------------------------------------ 2 (r s (c l s + 1) + l s) (s (w + p) + u) (%i3)
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p(t):=a*t^3+b*t^2+c*t+d; 3 2 (%o1) p(t) := a t + b t + c t + d (%i2) s:solve([p(-1)=p0,p(0)=p1,p(1)=p2,p(2)=p3],[a,b,c,d]); - p3 + 3 p2 - 3 p1 + p0 p2 - 2 p1 + p0 (%o2) [[a = - -----------------------, b = --------------, 6 2 p3 - 6 p2 + 3 p1 + 2 p0 c = - -----------------------, d = p1]] 6 (%i3) r(t):=-(-p3+3*p2-3*p1+p0)/6*t^3+(p2-2*p1+p0)/2*t^2-(p3-6*p2+3*p1+2*p0)/6*t+p1; - (- p3 + 3 p2 + (- 3) p1 + p0) 3 p2 - 2 p1 + p0 2 (%o3) r(t) := ------------------------------- t + -------------- t 6 2 - (p3 - 6 p2 + 3 p1 + 2 p0) + --------------------------- t + p1 6 (%i4) ratsimp(r(1/4)), expand; 5 p3 35 p2 105 p1 7 p0 (%o4) - ---- + ----- + ------ - ---- 128 128 128 128 (%i5) ratsimp(r(1/2)), expand; p3 9 p2 9 p1 p0 (%o5) - -- + ---- + ---- - -- 16 16 16 16 (%i6) ratsimp(r(3/4)), expand; 7 p3 105 p2 35 p1 5 p0 (%o6) - ---- + ------ + ----- - ---- 128 128 128 128 (%i7) factorout(expand(r(t)), 1-t, t); 3 3 3 3 2 2 p3 t p2 t p1 t p0 t p2 t p0 t p3 t (%o7) ----- - ----- + ----- - ----- + ----- + ----- - p1 (t - 1) (t + 1) - ---- 6 2 2 6 2 2 6 p1 t p0 t + p2 t - ---- - ---- 2 3 (%i8)
[assume,constant,cos,declare,diff,factorout,globalsolve,let,letsimp,lhs,posfun,realonly,sin,subst,trigsimp,true] [at,diff,factorout,sqrt] [constant,cos,declare,factorout,globalsolve,realonly,rhs,sin,true] [cos,eliminate,expand,factorout,sin] [expand,factor,factorout,ratsimp] [expand,factorout,ratsimp,solve] [factorout,sqrt] [factorout,subst] [factorout]
