LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=restart;with(DEtools):with(plots):with(student): First Order Equations Equation Separable or Bernoullide0_1:=diff(y(x),x)=y(x)-y(x)^2;sol:=dsolve(de0_1,y(x));soln:=dsolve({de0_1,y(0)=1/2},y(x));yy:=rhs(soln);
plot(yy,x=-10..15,thickness=4); Equation Separablede0_2:=9*y(x)*diff(y(x),x)+4*x=0;dsolve(de0_2,y(x));dsolve(de0_2,y(x),implicit); Equation Separable or Bernoullide0_3:=diff(y(x),x)+(x+1)*y(x)^3=0;dsolve( de0_3,y(x)); # the answer is a bit messydsolve(de0_3,y(x),implicit); # so we give an implicit solution# we can plot a solution usingDEplot(de0_3, y(x), x=0..20, [y(0)=2],numpoints=200,linecolor=black ); Equation First Order Linearde0_4:=diff(y(x),x)=3*(y(x)+1);dsolve( {de0_4,y(0)=1},y(x)); Equation Separable or Bernoullide0_5:=diff(y(x),x)=(1+2*x)*(1+y(x)^2);dsolve(de0_5 ,y(x)); Equation Separable de0_6:=diff(y(x),x)=exp(y(x)-x);sol:=dsolve( {de0_6,y(0)=0},y(x));sol:=dsolve(de0_6,y(x));Second and Higher Order Linearwith(VectorCalculus):W:=Wronskian([exp(x),cos(x),sin(x)],x);simplify(Determinant(W));Constant Coefficient 2nd Order Homogeneousde1:=diff(y(x),x$2)-4*y(x)=0;dsolve(de1,y(x));dsolve({de1,y(0)=2,D(y)(0)=0},y(x));de2:=diff(y(x),x$2)-12*diff(y(x),x)+36*y(x);dsolve(de2,y(x));Euler Cauchy Problemde3:=x^2*diff(y(x),x$2)+x*diff(y(x),x)=0;dsolve(de3,y(x));de3a:=x^2*diff(y(x),x$2)-x*diff(y(x),x)+2*y(x)=0;dsolve(de3a,y(x));Higher order equationde4:= diff(y(x),x$4)-y(x)=0;dsolve(de4,y(x));de4a:= diff(y(x),x$3)-2*diff(y(x),x$2)-diff(y(x),x)+2*y(x)=0;dsolve(de4a,y(x));Some nonhomogeneous Problemsde5:=diff(y(x),x$2)- diff(y(x),x)=2*exp(2*x);dsolve({de5,y(0)=2,D(y)(0)=2},y(x));de6:=diff(y(x),x$2)+ 3*diff(y(x),x)-4*y(x)=50*x*exp(x);dsolve(de6,y(x));de7B:=diff(y(x),x$2)+ 4*y(x)=10*exp(x);dsolve(de7B,y(x)); Laplace Transforms Compute some Laplace transformswith(inttrans):f1:=2*t^(3/2)+t^2*exp(t);F1:=laplace(f1,t,s); f2:= 2*t*sin(3*t);F2:= laplace(f2,t,s) ;f2B:=cos(2*t)*exp(-4*t) ;F2B:= laplace(f2B,t,s) ;expand(denom(F2B));f3:=exp(-t)*sin(2*t)*t;F3:=laplace(f3,t,s);factor(expand(denom(F3)));f:=1+Heaviside(t-1)*(t-1);F:=laplace(f,t,s);f:=1+Heaviside(t-3)*(t-1);F:=laplace(f,t,s);Compute some Inverse Laplace transformsF4:=(s-1)/(s*(s^2+1));f4:=invlaplace(F4,s,t);F5:=exp(-2*s)/((s-1)*(s^2-2*s+2));f5:=invlaplace(F5,s,t);F:=2*exp(-s)/(s^2+2*s+5);f:=invlaplace(F,s,t);Solve Differential Equation using Laplace transformsde7:=diff(y(t),t$2)+ diff(y(t),t)=Dirac(t-1);eqn:=laplace(de7,t,s);eqn := subs( [y(0) = 0, D(y)(0) = 0], eqn);Y1 := solve(eqn, laplace(y(t), t, s)) ;Y2 :=exp(-s)*convert(1/(s*(s+1)),parfrac,s);yy:=invlaplace(Y2,s,t);or we could simply type dsolve({de7,y(0)=0,D(y)(0)=0},y(t), method=laplace);de8:=diff(y(t),t$2)-y(t)=6*exp(2*t);dsolve({de8,y(0)=1,D(y)(0)=2},y(t),method=laplace);de9:=diff(y(t),t$2)+y(t)=2*Dirac(t-Pi);dsolve({de9,y(0)=1,D(y)(0)=1},y(t),method=laplace);Power Serieswith(powseries);Example 1: Power Series a:=powsolve(diff(y(x),x)-1/2*y(x)=0,y(0)=a_0);tp1:=tpsform(a,x,5);a(_k);tp2:=subs(a_0=1,tp1);dsolve({ diff(y(x),x)- 1/2*y(x)=0,y(0)=1},y(x));Tpoly:=convert(tp2,polynom);plot({Tpoly,exp(1/2*x)},x=-2..1);Example 2: Power Serisa:=powsolve((x-1)*diff(y(x),x)-3*y(x)=0,y(0)=1);tp3:=tpsform(a,x,3);a(_k);sol:=dsolve({(x-1)*diff(y(x),x)- 3*y(x)=0,y(0)=1},y(x));yy:=rhs(sol);tp4:=convert(tp3,polynom);plot({tp4,yy},x=-2..4);