*leon.magiera at wp.pl* leon.magiera at wp.pl
<maxima%40math.utexas.edu?Subject=Re%3A%20%5BMaxima%5D%20problems&In-Reply-To=%3C4f016750a81f33.02531046%40wp.pl%3E>
*Mon Jan 2 02:14:08 CST 2012 *write:
Hi
Problem 1
How, using Maxima, to solve in elegant way the equation
eq:sqrt(x+3-4*sqrt(x-1))+sqrt(x+8-6*sqrt(x-1))=1;
The solution is
5<=x<=10
Problem 2
ex:1/(sqrt((x-b)^2+a^2)*sqrt((x-b)^2/a^2+1));
integrate(ex,x);
Is the above integral too hard ?
Best
L.M.
********************************************************
Problem 1
(%i1) eq:sqrt(x+3-4*sqrt(x-1))+sqrt(x+8-6*sqrt(x-1))=1;
(%o1) sqrt(x-4*sqrt(x-1)+3)+sqrt(x-6*sqrt(x-1)+8)=1
The solution we can see from
(%i2) wxplot2d([lhs(eq)], [x,1,15])$
(%t2) << Graphics >>
(%i3) assume(t>0);
(%o3) [t>0]
(%i4) eq1:subst(x=1+t^2,eq);
(%o4) sqrt(t^2-4*t+4)+sqrt(t^2-6*t+9)=1
(%i5)
eq2:subst([t^2-4*t+4=factor(t^2-4*t+4),t^2-6*t+9=factor(t^2-6*t+9)],eq1);
(%o5) abs(t-2)+abs(t-3)=1
Next we analyse equation eq2 in cases:
a) t<2
(%i6) forget(t>0);assume(t<2);
(%o6) [t>0]
(%o7) [t<2]
(%i8) ratsimp(eq2);
(%o8) 5-2*t=1
(%i9) solve(%);
(%o9) [t=2]
Set of solutions is empty
b) 2<=t<=3
(%i10) forget(t<2)$ assume(t>=2,t<=3);
(%o11) [t>=2,t<=3]
(%i12) ratsimp(eq2);
(%o12) 1=1
Set of solutions is 2<=t<=3 or 5<=x<=10 ( because x=1+t^2)
c) t>3
(%i13) forget(t>=2,t<=3); assume(t>3);
(%o13) [t>=2,t<=3]
(%o14) [t>3]
(%i15) ratsimp(eq2);
(%o15) 2*t-5=1
(%i16) solve(%,t);
(%o16) [t=3]
Set of solutions is empty
problem 2
(%i17) ex:1/(sqrt((x-b)^2+a^2)*sqrt((x-b)^2/a^2+1));
(%o17) 1/(sqrt((x-b)^2+a^2)*sqrt((x-b)^2/a^2+1))
(%i18) radcan(%);
(%o18) abs(a)/(x^2-2*b*x+b^2+a^2)
(%i19) integrate(%,x);
(%o19) (abs(a)*atan((2*x-2*b)/(2*a)))/a
Best, Aleksas D