# Solve for the exact answers # System x'=3x-2y, y'=5x-4y, x(0)=3, y(0)=6 de1:=diff(x(t),t)=3*x(t)-2*y(t);de2:=diff(y(t),t)=5*x(t)-4*y(t);ic:=x(0)=3,y(0)=6; dsolve([de1,de2,ic],[x(t),y(t)]); Ans:=t->evalf(<2*exp(-2*t)+exp(t), 5*exp(-2*t)+exp(t)>); # Exact answers ; # Approximate methods are Euler, Heun, RK4. # Notation for t-values: t0=initial point, t1=t0+h, t2=t1+h, ... # Notation for u-values: 1u0=u(t0)=initial values, u1 approx u(t1), u2 approx u(t2), .... # Solve for approximate answers using Euler's Method F:=(t,u)-><3*u[1]-2*u[2],5*u[1]-4*u[2]>; h:=0.1; t0:=0; u0:=<3,6>; u1:=u0+h*F(t0,u0);t1:=t0+h;Ans(t1); u2:=u1+h*F(t1,u1);t2:=t1+h;Ans(t2); # Solve for approximate answers using Heun's Method h:=0.1; t0:=0; u0:=<3,6>; w:=u0+h*F(t0,u0);t1:=t0+h;u1:=u0+h*(F(t0,u0)+F(t1,w))/2;Ans(t1); w:=u1+h*F(t1,u1);t2:=t1+h;u2:=u1+h*(F(t1,u1)+F(t2,w))/2;Ans(t2); # Solve for approximate answers using the RK4 Method h:=0.1; t0:=0; u0:=<3,6>; k1:=h*F(t0,u0);t1:=t0+h; k2:=h*F(t0+h/2,u0+k1/2); k3:=h*F(t0+h/2,u0+k2/2); k4:=h*F(t1,u0+k3); u1:=u0+(k1+2*k2+2*k3+k4)/6;Ans(t1); k1:=h*F(t1,u1);t2:=t1+h; k2:=h*F(t1+h/2,u1+k1/2); k3:=h*F(t1+h/2,u1+k2/2); k4:=h*F(t1,u1+k3); u2:=u1+(k1+2*k2+2*k3+k4)/6;Ans(t2);