PKÏsrXñB–Hmimetypetext/x-wxmathmlPKÏsrXQdBV55 format.txt This file contains a wxMaxima session in the .wxmx format. .wxmx files are .xml-based files contained in a .zip container like .odt or .docx files. After changing their name to end in .zip the .xml and eventual bitmap files inside them can be extracted using any .zip file viewer. The reason why part of a .wxmx file still might still seem to make sense in a ordinary text viewer is that the text portion of .wxmx by default isn't compressed: The text is typically small and compressing it would mean that changing a single character would (with a high probability) change big parts of the whole contents of the compressed .zip archive. Even if version control tools like git and svn that remember all changes that were ever made to a file can handle binary files compression would make the changed part of the file bigger and therefore seriously reduce the efficiency of version control wxMaxima can be downloaded from https://github.com/wxMaxima-developers/wxmaxima. It also is part of the windows installer for maxima (https://wxmaxima-developers.github.io/wxmaxima/). If a .wxmx file is broken but the content.xml portion of the file can still be viewed using an text editor just save the xml's text as "content.xml" and try to open it using a recent version of wxMaxima. If it is valid XML (the XML header is intact, all opened tags are closed again, the text is saved with the text encoding "UTF8 without BOM" and the few special characters XML requires this for are properly escaped) chances are high that wxMaxima will be able to recover all code and text from the XML file. PKÏsrXcª2§8%8% content.xml Kedvenc Matematikai Kísérleteim Alapok Halmazok és függvények Számok Határérték Differenciálszámítás Integrálszámítás Lineáris algebra Többváltozós függvények Mérték és valószínűség Fourier-elmélet Komplex függvénytan Közönséges differenciálegyenletek Alapfogalmak Egyváltozós variációszámítás Elemi megoldási módszerek Megjegyzés Egzakt egyenletek Integráló szorzó Feladat (1) kill(all); (%o0) done load("ode2"); (%o1) /usr/share/maxima/5.43.2/share/diffequations/ode2.mac depends(μ,t); (%o2) [μ

t

]
ft:(t+t^2*(t^2+x^2)); fx:x; (%o3) t2*

x2+t2

+t(%o4) x
diff(ft*μ,x); diff(fx*μ,t); (%o5) 2*t2*x*μ(%o6) x*

dd*t*μ

ode2('diff(μ,t)=2*t^2*μ,μ,t); (%o7) μ=%c*%e2*t33 subst(%c=1,%); (%o8) μ=%e2*t33 μ:rhs(%); (%o9) %e2*t33 0; gt:ft*μ; gx:fx*μ; (%o30) 0(%o31) %e2*t33*

t2*

x2+t2

+t

(%o32) %e2*t33*x
0; diff(gt,x); diff(gx,t); (%o22) 0(%o23) 2*t2*%e2*t33*x(%o24) 2*t2*%e2*t33*x 0; ht:subst(x=0,gt); (%o33) 0(%o34)

t4+t

*%e2*t33
0;hx:subst(x=y,gx); (%o35) 0(%o36) %e2*t33*y 0;F:integrate(ht,t)+integrate(hx,y,0,x); (%o37) 0(%o38) %e2*t33*x22+323*gamma_incomplete

53,−2*t33

253
−gamma_incomplete

23,−2*t33

223*313
0;diff(F,t); (%o40) 0(%o41) t2*%e2*t33*x2+t4*%e2*t33+t*%e2*t33 0;diff(F,x); (%o42) 0(%o43) %e2*t33*x (2) etc. Feladat Feladat Feladat Szeparábilis egyenletek Feladat Feladat (1) y:k*C/24*x^4+c3*x^3+c2*x^2+c1*x+c0; (%o11) C*k*x424+c3*x3+c2*x2+c1*x+c0 e1:subst(x=0,y)=0; (%o23) c0=0 e2:subst(x=L,y)=0; (%o24) C*L4*k24+L3*c3+L2*c2+L*c1+c0=0 ypp:diff(y,x,2); (%o19) C*k*x22+6*c3*x+2*c2 e3:subst(x=0,ypp)=0; (%o25) 2*c2=0 e4:subst(x=L,ypp)=0; (%o26) C*L2*k2+6*L*c3+2*c2=0 solve([e1,e2,e3,e4],[c1,c2,c3,c4]); (%o27) [] subst(c0=0,y); (%o28) C*k*x424+c3*x3+c2*x2+c1*x subst(c2=0,y); (%o29) C*k*x424+c3*x3+c1*x+c0 subst() (2) etc. Feladat (1) szep. kill(all); (%o0) done eq:1+x^2+t*x*'diff(x,t)=0; (%o2) t*x*

dd*t*x

+x2+1=0
ode2(eq,x,t); (%o3) −log

x2+1

2
=log

t

+%c
(2) szep. eq:t*x*(1+t^2)*'diff(x,t)=1+x^2; (%o30) t*

t2+1

*x*

dd*t*x

=x2+1
0;ode2(eq,x,t); (%o32) 0(%o33) log

x2+1

2
=%c−log

t2+1

−2*log

t

2
etc. Feladat (1) hom. kill(all); (%o0) done eq:'diff(x,t)=(t+x)/(t+2*x); (%o1) dd*t*x=x+t2*x+t 0;s:ode2(eq,x,t); (%o7) 0(%o8) %c*t=%e−log

2*x−2*t2*x+2*t

+2*log

2*x2−t2t2

232
sprint(ode2(eq,x,t))$ %c*t = %e^−((log((2*x−sqrt(2)*t)/(2*x+sqrt(2)*t))+sqrt(2)*log((2*x^2−t^2)/t^2))/2^(3/2)) (2) etc. (3) etc. (4) etc. (5) etc. (6) kill(all); (%o0) done eq:x*(x-2*y)*'diff(y,x)=y*(y-2*x); (%o1) x*

x−2*y

*

dd*x*y

=y*

y−2*x

ode2(eq,y,x); (%o2) x2*y−x*y2=%c subst('diff(u,x)*x+u,'diff(y,x),eq); (%o3) x*

dd*x*u

*x+u

*

x−2*y

=y*

y−2*x

subst(u*x,y,%); (%o4) x*

x−2*u*x

*

dd*x*u

*x+u

=u*x*

u*x−2*x

%/x^2; (%o5)

x−2*u*x

*

dd*x*u

*x+u

x
=u*

u*x−2*x

x
eq1:ratsimp(%); (%o6)

1−2*u

*

dd*x*u

*x−2*u2+u=u2−2*u
ode2(eq1,u,x); (%o7) −log

u

+log

u−1

3
=log

x

+%c
(7) kill(all); (%o0) done eq:(x^2+y^2)*'diff(y,x)=x*y; (%o1)

y2+x2

*

dd*x*y

=x*y
ode2(eq,y,x); (%o2) 2*y2*log

y

−x2
2*y2
=%c
Feladat kill(all); (%o0) done (1) eq:'diff(x,t)=(t-x)^2+1; (%o1) dd*t*x=

t−x

2
+1
ode2(eq,x,t); (%o2) false y:x-t; (%o3) x−t depends(x,t); (%o5) [x

t

]
diff(y,t); (%o6) dd*t*x−1 eq1:'diff(y,t)=y^2; (%o7) dd*t*

x−t

=

x−t

2
ode2(eq1,y,t); (%o8) −1x−t=t+%c etc. Feladat (1) kill(all); (%o0) done eq:(x+y+1)+(2*x+2*y+1)*'diff(y,x)=0; (%o6)

2*y+2*x+1

*

dd*x*y

+y+x+1=0
ode2(eq,y,x); (%o7) false u:x+y; (%o8) y+x depends(y,x); (%o9) [y

x

]
diff(u,x); (%o10) dd*x*y+1 eq1:('diff(u,x)-1)*(2*x+2*y+1)+y+x+1=0; (%o11)

2*y+2*x+1

*

dd*x*

y+x

−1

+y+x+1=0
ode2(eq1,u,x); (%o12) y+x=−log

2*y+2*x+1

4
+x2+%c
etc. Feladat kill(all); (%o0) done eq:'diff(v,t)=k*v; (%o1) dd*t*v=k*v ode2(eq,v,t); (%o2) v=%c*%ek*t subst(1.5,%c,%); (%o3) v=1.5*%ek*t subst([t=4,v=1],%); (%o4) 1=1.5*%e4*k solve(%,k); rat: replaced -1.5 by -3/2 = -1.5(%o5) [k=log

214*%i314

,k=log

−214314

,k=log

−214*%i314

,k=log

214314

]
float(%), numer; (%o6) [k=1.570796326794896*%i−0.1013662770270411,k=3.141592653589793*%i−0.1013662770270411,k=−1.570796326794896*%i−0.1013662770270411,k=−0.1013662770270411] %[4]; (%o7) k=−0.1013662770270411 subst([k=-log(2/3)/4],%o3); (%o8) v=1.5*%e−log

23

*t
4
integrate(%,t,0,inf); defint: integral is divergent. -- an error. To debug this try: debugmode(true); etc. Feladat Feladat Feladat Feladat Feladat Lineáris egyenletek Feladat (1) kill(all); (%o0) done ode2(t*'diff(x,t)+2*x=3*t,x,t); (%o4) x=t3+%ct2 ic1(%,t=0,x=0); expt: undefined: 0 to a negative exponent.#0: ic1(soln=x = (t^3+%c)/t^2,xc=t = 0,yc=x = 0) -- an error. To debug this try: debugmode(true); subst([%c=0],%o4); (%o6) x=t etc. Feladat kill(all); (%o0) done (1) eq:'diff(x,t)+x=-1/x; (%o2) dd*t*x+x=−1x ode2(eq,x,t); (%o6) −log

x2+1

2
=t+%c
etc. Feladat (1) kill(all); (%o0) done e:2*'diff(y,x)-3*y/x=3*x*y^(1/2); (%o1) 2*

dd*x*y

−3*yx=3*x*y
ode2(e,y,x); (%o10) y=

3*%e5*log

x

4
5+%c

2*%e3*log

x

2
radcan(%); (%o11) y=9*x4+30*%c*x114+25*%c2*x3225 (2) kill(all); (%o0) done e:y-'diff(y,x)=(1+x)*y^2; (%o1) y−dd*x*y=

x+1

*y2
ode2(e,y,x); (%o2) y=%ex

x−1

*%ex+%ex+%c
expand(%); (%o3) y=%exx*%ex+%c etc. Feladat Feladat Feladat Feladat Feladat Feladat Feladat Feladat Feladat kill(all); (%o0) done depends(y,x); (%o1) [y

x

]
diff(y^2,x); (%o2) 2*y*

dd*x*y

eq:-y/'diff(x,y)=c; (%o3) −ydd*y*x=c ode2(eq,x,y); (%t4) −ydd*y*x=cfirst order equation not linear in y'(%o4) false eq1:'diff(x,y)=-c/y; (%o5) dd*y*x=−cy ode2(eq1,x,y); (%o6) x=%c−c*log

y

etc. Feladat kill(all); (%o0) done e:[r*cos(φ),r*sin(φ),a*acosh(r/a)]; (%o1) [r*cos

φ

,r*sin

φ

,a*acosh

ra

]
ek:diff(e,φ); (%o2) [−r*sin

φ

,r*cos

φ

,0]
ekm:matrix(ek); (%o3) −r*sin

φ

r*cos

φ

0
ekn2:ekm . transpose(ekm); (%o4) r2*sin

φ

2
+r2*cos

φ

2
depends(r,φ); (%o5) [r

φ

]
eg:diff(e,φ); (%o6) [

dd*φ*r

*cos

φ

−r*sin

φ

,

dd*φ*r

*sin

φ

+r*cos

φ

,dd*φ*rr2a2−1]
egm:matrix(eg); (%o7)

dd*φ*r

*cos

φ

−r*sin

φ

dd*φ*r

*sin

φ

+r*cos

φ

dd*φ*rr2a2−1
egn2:egm . transpose(egm); (%o9)

dd*φ*r

*sin

φ

+r*cos

φ

2
+

dd*φ*r

*cos

φ

−r*sin

φ

2
+

dd*φ*r

2
r2a2−1
expand(%); (%o10)

dd*φ*r

2
*sin

φ

2
+r2*sin

φ

2
+

dd*φ*r

2
*cos

φ

2
+r2*cos

φ

2
+

dd*φ*r

2
r2a2−1
ratsimp(%); (%o11)

r2−a2

*

dd*φ*r

2
+r4−a2*r2

*sin

φ

2
+

r2−a2

*

dd*φ*r

2
+r4−a2*r2

*cos

φ

2
+a2*

dd*φ*r

2
r2−a2
egn2:trigsimp(%); (%o12) r2*

dd*φ*r

2
+r4−a2*r2
r2−a2
egm . transpose(ekm); (%o13) r*cos

φ

*

dd*φ*r

*sin

φ

+r*cos

φ

−r*sin

φ

*

dd*φ*r

*cos

φ

−r*sin

φ

ip2:%^2; (%o14)

r*cos

φ

*

dd*φ*r

*sin

φ

+r*cos

φ

−r*sin

φ

*

dd*φ*r

*cos

φ

−r*sin

φ

2
expand(%); (%o15) r4*sin

φ

4
+2*r4*cos

φ

2
*sin

φ

2
+r4*cos

φ

4
ee:%=egn2*ekn2/2; (%o16) r4*sin

φ

4
+2*r4*cos

φ

2
*sin

φ

2
+r4*cos

φ

4
=

r2*

dd*φ*r

2+r4−a2*r2

*

r2*sin

φ

2+r2*cos

φ

2

2*

r2−a2

ee1:trigsimp(%); (%o17) r4=r4*

dd*φ*r

2
+r6−a2*r4
2*r2−2*a2
ee2:ee1*(2*r^2-2*a^2)/r^4; (%o18) 2*r2−2*a2=r4*

dd*φ*r

2
+r6−a2*r4
r4
ratsimp(%); (%o19) 2*r2−2*a2=

dd*φ*r

2
+r2−a2
ee3:%-(r^2-a^2); (%o20) r2−a2=

dd*φ*r

2
ee4:sqrt(%); (%o21) r2−a2=dd*φ*r de1:'diff(r,φ)=sqrt(r^2-a^2); (%o22) dd*φ*r=r2−a2 ode2(%,r,φ); (%o23) log

2*r2−a2+2*r

=φ+%c
de2:'diff(r,φ)=-sqrt(r^2-a^2); (%o24) dd*φ*r=−r2−a2 ode2(%,r,φ); (%o25) −log

2*r2−a2+2*r

=φ+%c
Implicit egyenletek Feladat kill(all); (%o0) done eq:t=x*'diff(x,t)/(1+'diff(x,t)^2); (%o1) t=x*

dd*t*x

dd*t*x

2
+1
ode2(eq,x,t); (%t2) t=x*

dd*t*x

dd*t*x

2
+1
first order equation not linear in y'(%o2) false
eq1:t=x*p/(1+p^2); (%o8) t=p*xp2+1 depends(p,x); (%o9) [p

x

]
eq2:1/p=diff(rhs(eq1),x); (%o13) 1p=

dd*x*p

*x
p2+1
−2*p2*

dd*x*p

*x

p2+1

2
+pp2+1
eq3:eq2*(p*(p^2+1)); (%o19) p2+1=p*

p2+1

*

dd*x*p

*xp2+1−2*p2*

dd*x*p

*x

p2+1

2
+pp2+1

eq4:ratsimp(%); (%o21) p2+1=−

p3−p

*

dd*x*p

*x−p4−p2
p2+1
eq4*(p^2+1); (%o22)

p2+1

2
=−

p3−p

*

dd*x*p

*x+p4+p2
ratsimp(%); (%o23) p4+2*p2+1=

p−p3

*

dd*x*p

*x+p4+p2
ode2(%,p,x); (%o24) 2*log

p2+1

−p2
2
=log

x

+%c
etc. Magasabbrendű egyenletek Feladat kill(all); (%o0) done integrate(integrate(sin(t)/t,t),t); (%o1) −%i*gamma_incomplete

0,%i*t

*t−%i*gamma_incomplete

0,−%i*t

*t−gamma_incomplete

1,%i*t

−gamma_incomplete

1,−%i*t

2
etc. Feladat kill(all); (%o0) done eq:t*'diff(x,t,2)=2*x*'diff(x,t); (%o2) t*

d2d*t2*x

=2*x*

dd*t*x

ode2(eq,x,t); (%o3) false eq1:t*'diff(x,t,2)='diff(x^2,t); (%o4) t*

d2d*t2*x

=dd*t*x2
ode2(eq1,x,t); (%o5) false depends(x,t); (%o6) [x

t

]
diff(t*x,t,2); (%o7) t*

d2d*t2*x

+2*

dd*t*x

eq2:'diff(t*x,t,2)-2*'diff(x,t)='diff(x^2,t); (%o8) d2d*t2*

t*x

−2*

dd*t*x

=dd*t*x2
eq3:'diff(t*x,t)-2*x=x^2+c; (%o10) dd*t*

t*x

−2*x=x2+c
ode2(eq3,x,t); (%t12) dd*t*

t*x

−2*x=x2+cnot a proper differential equation(%o12) false
diff(t*x,t); (%o13) t*

dd*t*x

+x
eq4:%-2*x=x^2+c; (%o14) t*

dd*t*x

−x=x2+c
ode2(eq4,x,t); Is c−0.25 positive or negative? Maxima encountered a Lisp error: SIMPLE-ERROR: Console interrupt.Automatically continuing.To enable the Lisp debugger set *debugger-hook* to nil. assume(c-1/4>0); (%o17) [c>14] ode2(eq4,x,t); (%o18) 2*atan

2*x+14*c−1

4*c−1
=log

t

+%c
Feladat Vegyes módszerek Operátorszámítás Feladat
PKÏsrXñB–HmimetypePKÏsrXQdBV55 5format.txtPKÏsrXcª2§8%8% ’content.xmlPK§ó+