Author: Frank Postel Date : 07. Aug. 1995 Format: dvi, postscript
In this note it is demonstrated how MuPAD 1.3 solves 33 problems of the testsuite and which problems MuPAD can't solve.
Initialize this part by loading the linalg library into MuPAD:
loadlib("linalg");
>> MS:= SquareMatrix( 30,Rational ): H := MS( func(1/(i+j-1),i,j) ): >> iH:= H^(-1): iH[30,30]; 53338390370510951379337308012614400
+- -+ | a, 1, 0 | | 1, 1, 1 | | 1, 1, 2 | +- -+ >> MExpr:= Matrix( ExpressionField(normal) ): A:= MExpr( [[a,1,0],[1,1,1],[1,1,2]] ); +- -+ | a, 1, 0 | | | | 1, 1, 1 | | | | 1, 1, 2 | +- -+ >> le:= linalg::eigenValues( A ):The following outputs were cutted due to the long results:
>> le[1]; / / 2 3 | 2 | a a | - 3 a + a + 9 | - -- + -- + ... \ \ 6 27 1/2 2 3 4 1/2 \ 1/3 \ 3 (108 a - 90 a + 32 a - 5 a - 81) | | ... + ------------------------------------------- + 1/2 | | 18 / / >> le[2]; / | 2 1/2 1/2 2 1/2 | 3 a - a + (- 9 I) 3 + (3 I) a 3 + (- I) a 3 + ... \ 1/2 2 3 4 1/2 \ 1/3 \ 3 (108 a - 90 a + 32 a - 5 a - 81) | | ... + ------------------------------------------- + 1/2 | | 18 / / >> le[3]; / | 2 1/2 1/2 2 1/2 | 3 a - a + (9 I) 3 + (- 3 I) a 3 + (I) a 3 + ... \ 1/2 2 3 4 1/2 \ 1/3 \ 3 (108 a - 90 a + 32 a - 5 a - 81) | | ... + ------------------------------------------- + 1/2 | | 18 / /
>> radsimp( sqrt( 9+4*sqrt(5) ) ); 1/2 5 + 2
>> e:= (sin(2*x) - sin(x)*cos(x))/(cos(x)*(1+tan(x)^2)): combine( normal(expand(e)), sincos ); sin(x) sin(3 x) ------ + -------- 4 4
>> simplify( (exp(x) - 1)/(exp(x/2)+1),exp ); / x \ exp| - | - 1 \ 2 /
>> solve( x^4 + a*x^2 + 1 = 0, x); -- 1/2 2 1/2 1/2 1/2 2 1/2 1/2 | 2 (- a + (a - 4) ) 2 (- a + (a - 4) ) | ---------------------------, - ---------------------------, -- 2 2 1/2 2 1/2 1/2 1/2 2 1/2 1/2 -- 2 (- a - (a - 4) ) 2 (- a - (a - 4) ) | ---------------------------, - --------------------------- | 2 2 --
>> solve( 2*sin(x) + 5*cos(x) = 3, x ); -- / 1/2 \ / 1/2 \ -- | | 5 | | 5 | | | 2 PI k1 + 2 atan| - ---- + 1/4 |, 2 PI k2 + 2 atan| ---- + 1/4 | | -- \ 4 / \ 4 / --
>> solve( exp(2*x) + exp(x) - 1 = 0, x ); -- / 1/2 \ / 1/2 \ -- | | 5 | | 5 | | | 2 I PI k3 + ln| - ---- - 1/2 |, 2 I PI k4 + ln| ---- - 1/2 | | -- \ 2 / \ 2 / --
x*y^2 + 2*c + x*z = 1 x*y - x^2 + 2*z = 0 x*y - z = 0
>> solve( {x*y^2+2*c+x*z=1, x*y-x^2+2*z=0, x*y-z=0}, {x,y,z} ); -- -- 2 -- -- | | x x 3 | | | | y = -, z = --, x = RootOf(18 c + 4 x - 9, x) | | -- -- 3 3 -- --
>> s:= series( cosh(x)/sinh(x)-1/x,x,5 ); x 3 - + O(x ) 3 >> testtype( s,Type::Series( Taylor ) ); TRUE
Unable to do in MuPAD 1.3
>> sum( 1/(k^2+a^2), k=1..infinity ); / | 2 1/2 2 1/2 2 1/2 | psi((- a ) + 1) - psi(- (- a ) + 1) + 2 (- a ) | \ / 2 1/2 2 1/2 \ \ | - psi(k + (- a ) ) + psi(k - (- a ) ) | | limit| -----------------------------------------, k = infinity | | / | 2 1/2 | | \ 2 (- a ) / / 2 1/2 (2 (- a ) )
>> limit( asinh((u+a)/b) - asinh((u-a)/b), u=infinity ); 0
>> limit( exp(1/x^2)/(exp(1/x^2)+exp(1/x^4)) , x=0 ); 0
>> diff( (-1/5*x^2 + 1/5*x^2*tan(1/2*ln(x))^2 + 4/5*x^2*tan(1/2*ln(x))) / (1 + tan(1/2*ln(x))^2) , x ); / / ln(x) \ / ln(x) \2 | 8 x tan| ----- | 2 x tan| ----- | | 2 x \ 2 / \ 2 / | - --- + ---------------- + ----------------- + ... | 5 5 5 | \ / 2 / ln(x) \ 2 / ln(x) \2 \ \ | 2 4 x tan| ----- | x tan| ----- | | | | x \ 2 / \ 2 / | | ... | - -- + ----------------- + ---------------- | | \ 5 5 5 / / / / ln(x) \2 / / ln(x) \2 \2 \ / | x cos| ----- | | tan| ----- | + 1 | | \ \ 2 / \ \ 2 / / /The output was cutted due to the long result.
Unable to do in MuPAD 1.3
>> int( (2*x^4 + 1)/((x^5+x)*sqrt(x^4+1)) , x ); / 4 \ | 2 x + 1 | int| --------------------, x | | 5 4 1/2 | \ (x + x ) (x + 1) /As one can see, by now the integrator of MuPAD is not able to handle certain algebraic dependences of functions, but we hope that this will be implemented in a next MuPAD release.
>> int( exp(-x)*sin(x)*cos(x) , x ); cos(2 x) sin(2 x) - -------- - --------- 5 exp(x) 10 exp(x)
>> int( x^3/(exp(x)+1), x=0..infinity ); 4 7 PI ----- 120
>> int( ln(x)^2/sqrt(1-x^2), x=0..1 ); 3 2 PI PI ln(2) --- + --------- 24 2
>> laplace( sin(t), t, s ); 1 ------ 2 s + 1
>> ilaplace( %, s, t ); sin(t)
>> laplace( heaviside(t-2) * sin(t), t, s ); / cos(2) s sin(2) \ exp(- 2 s) | ------ + -------- | | 2 2 | \ s + 1 s + 1 /
>> ilaplace( %, s, t ); heaviside(t - 2) (cos(2) sin(t - 2) + sin(2) cos(t - 2))Use combine to get back the original function:
>> combine(%,sincos); sin(t) heaviside(t - 2)
>> fourier( exp(-t^2),t,x ); / 2 \ 1/2 | x | PI exp| - -- | \ 4 /
>> ifourier( exp(-t^2),t,x ); / 2 \ | x | exp| - -- | \ 4 / ----------- 1/2 2 PI
loadlib("ode");
>> solve( ode( x^2*diff(y(x),x)-x*y(x) = x*ln(x) , y(x) ) ); [C1 x - ln(x) - 1]
>> solve( ode(diff(y(x),x) - y(x)^2 - 3*y(x) + 4 = 0, y(x)) ); -- / ln(y - 1) ln(y + 4) \ -- | -4, 1, solve| - C2 - x + --------- - --------- = 0, y | | -- \ 5 5 / --
>> solve( ode( diff(y(x),x$2) - 4*y(x) = sin(x) , y(x)) ); -- sin(x) -- | - ------ + C3 exp(2 x) + C4 exp(- 2 x) | -- 5 --
>> solve( ode( diff(y(x),x$2) - 2/x*diff(y(x),x) + 3/x^2*y(x) = 2*x - 1, y(x) ) ); -- 3 / 1/2 \ / 1/2 \ -- | 2 2 x 3/2 | ln(x) 3 | 3/2 | ln(x) 3 | | | - x + ---- + C2 x cos| ---------- | + C3 x sin| ---------- | | -- 3 \ 2 / \ 2 / --
>> solve( ode( 8*diff(y(x),x$2) + 9*diff(y(x),x)^4 = 0, y(x) ) ); -- / / 9 y \1/2 / 16 C5 2 y \ \ -- | C6, solve| | - 2 C5 + --- | | - ----- + --- | = x + C7, y | | -- \ \ 4 / \ 27 3 / / --
>> solve( ode( 4*diff(y(x),x$4) - 12*diff(y(x),x$3) + 11*diff(y(x),x$2) - 3*diff(y(x),x) = 4*cos(x) , y(x) ) ); -- 14 cos(x) 18 sin(x) / 3 x \ | C8 - --------- + --------- + C10 exp(x) + C9 exp| --- | -- 65 65 \ 2 / / x \ -- + C11 exp| - | | \ 2 / --
x' - x + y' + 2 y = 1 + e^t y' + 2 y + z' + z = 2 + e^t x' - x + z' + z = 3 + e^t
>> solve( ode( { diff(x(t),t) - x(t) + diff(y(t),t) + 2*y(t) = 1+exp(t), diff(y(t),t) + 2*y(t) + diff(z(t),t) + z(t) = 2+exp(t), diff(x(t),t) - x(t) + diff(z(t),t) + z(t) = 3+exp(t) }, {x(t), y(t), z(t)} ) ); -- -- t exp(t) | | x(t) = -------- + C12 exp(t) - 1, | | 2 -- -- exp(t) C13 exp(t) C14 -- -- y(t) = ------ + -------, z(t) = ------ + ------ + 2 | | 6 2 4 exp(t) | | exp(t) -- --
Unable to do in MuPAD 1.3In the current version of MuPAD, the plot command can not handle singularities, therefore a plot of this function would only be possible for the interval (0,5].
>> implicitplot( fun(args(1)*exp(args(2))-args(2)*exp(-args(1)) - 1) , -4..4, -4..4, 7 ); show image
>> plot3d( Scaling=UnConstrained, [ Mode=Surface,[u, v, u^2*cos(v) + sin(u)], u=[-4.0,4.0],v=[-4.0,4.0], Color=[Height,[0.996109,0.164050,0.0], [0.992203,0.996109,0.0]], Style=[ColorPatches,AndMesh] ] ); show image
Unable to do in MuPAD 1.3
>> fieldplot( Axes = Origin, [ [-y^2, x^2], x = [-4, 4], y = [-4, 4], Grid = [30, 30] ] ); show image