# SECTION 1.2# Example 1phi := x^2-x^(-1);

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de := diff(y(x),x,x) - 2/x^2*y(x) = 0;

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subs(y(x)=phi,de);

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simplify(%);

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# Example 2phi := c1*exp(-x) + c2*exp(2*x);

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de := diff(y(x),x,x) - diff(y(x),x) - 2*y(x) = 0;

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subs(y(x)=phi,de);

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simplify(%);

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# Example 3phi := 'phi';

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eq := phi^2 - x^3 + 8 = 0;

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solve(eq,phi);

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phi := %[1];

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de := diff(y(x),x) = 3*x^2/(2*y(x));

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subs(y(x) = phi,de);

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simplify(%);

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# Example 4eq := x + y(x) + exp(x*y(x)) = 0;

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diff(eq,x);

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expand(%);

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# The above equation is easily seen to be equivalent to the original differential equation. # Example 5eq := 4*x^2 - y(x)^2 = C;

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diff(eq,x);

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with(plots):# Review syntax of contourplot:?contourplotcontourplot(lhs(eq),x=-2..2,y=-4..4,scaling=constrained,contours=[-4,-1,0,1,4]);

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***********47$F/7$FhasFgdm7$FhbsF]as7$Fabs7$$!3#eB)eqk<w6F/$!33IN#)eqk(3$F/7$7$$!3ddG9dG9U6F/FfnF[cs7$7$Fbcs$!3#4++++++/$F/7$$!3Z++++vo%3"F/$!3]++++]i]HF/7$7$Fj[l$!3i666666")GF/Fhcs7$7$Fj[l$"3H566666")GF/7$FbcsF_q7$FedsFebs7$F^ds7$$!3Srmmmmm;**F[[l$!3Knmmmmm;GF/7$7$$!3?qmmmmm">*F[[lFboFhds7$F^es7$$!3TZr&G9dy'*)F[[l$!3$=dG9dGko#F/7$7$$!3g0++++++))F[[l$!3y+++++DhEF/Fbes7$7$Fj\n$"3,*********\7m#F/7$F_esFep7$FafsFbds7$7$Fj\nF[fs7$$!3g]Q:YQ:'*zF[[l$!3!=Bp2Bp2c#F/7$7$Ff_n$!3g+++++DhCF/Fefs7$7$Fies$"3X*********\7m#F/7$$!3%>nmmmmm,)F[[l$"3eKLLLLLjDF/7$7$Ff_n$"3r*********\7Y#F/Fbgs7$7$Ff_n$!30,++++DhCF/7$$!3oOLLLL$e*pF[[l$!3&QLLLLL3W#F/7$7$$!3"R+++++ve'F[[lF_rF_hs7$7$Ffhs$!3O+++++++CF/7$$!3x744444ffF[[l$!3->=====GBF/7$7$F]bn$!3"y&G9dG9(G#F/F\is7$7$F]bn$"3[cG9dG9(G#F/7$FfhsFds7$FiisFhgs7$Fbis7$$!3y.+++++v[F[[l$!3)4+++++]A#F/7$7$Ff]p$!3!3++++++:#F/F\js7$Ffis7$$!3#Q+++++Dh%F[[l$"3C**********\-AF/7$7$Ff]p$"3Z************\@F/Ffjs7$Fbjs7$$!3(4yxxxxxs$F[[l$!3[XWWWWWM@F/7$7$$!3d.+++++vFF[[lFbtF`[t7$Ff[t7$Fhiq$!39,++++Dh?F/7$7$Fc^p$!30,+++++b?F/Fj[t7$7$$!3!Q++++++S#F[[l$"3s***********\0#F/7$Fg[tFeu7$Fg\tF\[t7$7$Fc\tF_\t7$$!3#*)G9dG9d8"F[[l$!3zVr&G9dG,#F/7$7$F`_p$!3#ymmmmm;+#F/F[]t7$7$Fc^p$"3F***********\0#F/7$$!3Q.+++++]AF[[l$"3Q************\?F/7$7$F`_p$"3/mmmmmm,?F/Fh]t7$Fa]t7$$"3M,LLLLL$3%F`_lFb]t7$7$Fh_pFb]tFb^t7$F^^t7$$!3tqLLLLL$3%F`_lF_^t7$7$Fh_pF_^tFh^t7$Ff^t7$$"3W'***********\AF[[l$!3r++++++]?F/7$7$Fc`pF_\tF^_t7$F\_t7$$"3)>G9dG9d8"F[[l$"3YUr&G9dG,#F/7$7$$"3'o************R#F[[lFe\tFf_t7$7$FaapFcjs7$$"3"p**********\x#F[[lFbt7$Fa`t7$F]`tF_\t7$7$Fc`pFf]t7$F_^r$"3!)*********\71#F/7$7$Fb`tFeuFh`t7$F\at7$$"3'[xxxxxxs$F[[l$"3:WWWWWWM@F/7$7$FaapF][tF^at7$F``t7$$"3;(*********\7YF[[l$!3,,++++]-AF/7$7$F`enFcisFfat7$Fdat7$$"37(**********\([F[[l$"3k***********\A#F/7$7$F`enFgisF^bt7$7$FggnF\gs7$$"3D(*********\(e'F[[lFjhs7$7$FhbtF_rF\bt7$Fdbt7$$"36144444ffF[[l$"3p<=====GBF/7$7$FhbtFdsF]ct7$Fcct7$$"3+ILLLL$e*pF[[l$"3'HLLLLL3W#F/7$7$FggnFigsFect7$7$FggnF]hs7$$"3Glmmmmm;!)F[[l$!3NMLLLLLjDF/7$7$F`inF[fsF^dt7$F[dt7$$"3%R%Q:YQ:'*zF[[l$"3-I#p2Bp2c#F/7$7$F`inF_fsFfdt7$7$Fe`lF_ds7$$"3kkmmmmm">*F[[lFbo7$F_etFddt7$7$F`inF`gs7$$"3vSr&G9dy'*)F[[l$"3%4dG9dGko#F/7$7$F`etFepFeet7$F[ft7$$"3&emmmmmm"**F[[l$"3*fmmmmmm"GF/7$7$Fe`lFcdsF]ft7$7$F`clFbbs7$$"3"p&G9dG9U6F/Ffcs7$7$FgftFfnF^et7$Fcft7$$"3!)*******\(o%3"F/$"3;********\i]HF/7$7$FgftF_qF\gt7$Fbgt7$$"3$\B)eqk<w6F/$"3vGN#)eqk(3$F/7$7$F`clFfbsFdgt7$7$FjelFd`s7$$"3a********\P\8F/F97$F]htFeft7$Fjgt7$$"3d))))))))))Qm7F/$"3L@AAAAAFKF/7$7$F^htFgdmFbht7$Fhht7$$"3Ct%*y:j_b8F/$"3O^5Uot%*oLF/7$7$FjelF^asFjht7$F\ht7$$"3k**********\-9F/$!3G++++++XMF/7$7$FhhlFd_sFbit7$F`it7$Fd[o$"3&))*********\7NF/7$7$FhhlFh_sFjit7$7$$"3r************z;F/Fi]s7$$"3V**********\V:F/Ffjp7$7$$"3l**********\V:F/FfjpFhit7$F^jt7$$"3xv/>w/>J:F/$"3WY!>w/>wl$F/7$7$FdjtFhbmF[[u7$7$FhjtFhbm7$Fc\o$"3'y!44444/QF/7$7$FajtF]^sFd[u7$7$$"3![aaaaa/t"F/FcfoF`jt7$Fh[u7$$"313EyM/8/<F/$"34"yM/8R<&RF/7$7$$"3-XXXXXXI<F/FdgoF^\u-F^`m6&F``mFa`mFg`mFg`m-%+AXESLABELSG6$Q"x6"Q"yF]]u-%*LINESTYLEG6#F,-F^`m6#%%NONEG-%+PROJECTIONG6#Fa`m-%(SCALINGG6#%,CONSTRAINEDG-%,ORIENTATIONG6$$"#XF,F_^u-%*GRIDSTYLEG6#%,RECTANGULARG

#Example 6phi := sin(x) - cos(x);

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de := diff(y(x),x,x) + y(x) = 0;

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subs(y(x)=phi,de);

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simplify(%);

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subs(x=0,phi);

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simplify(%);

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phiprime := diff(phi,x);

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subs(x=0,phiprime);

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simplify(%);

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# Example 7phi := c1*exp(-x) + c2*exp(2*x);

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phiprime := diff(phi,x);

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subs(x=0,phi) = 2;

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ic1 := simplify(%);

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subs(x=0,phiprime) = -3;

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ic2 := simplify(%);

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solve({ic1,ic2},{c1,c2});

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subs(%,phi);

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# Example 8f := x^2 - x*y^3;

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diff(f,y);

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# Example 9f := 3*y^(2/3);

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diff(f,y);

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