RKF45Kinda.nb

(*Define the system of ODEs*)
system = {y1'[t] == y2[t], y2'[t] == -0.5*y1[t], y3'[t] == y4[t], 
   y4'[t] == -0.5*y3[t]};

(*Initial conditions*)
initialConditions = {y1[0] == 1, y2[0] == 0, y3[0] == 0, y4[0] == 1};

(*Initial and final time*)
t0 = 0;
tf = 1;

(*Solve the system of ODEs*)
sol = NDSolve[{system, initialConditions}, {y1, y2, y3, y4}, {t, t0, 
    tf}];

(*Plot the solutions*)
Plot[Evaluate[{y1[t], y2[t], y3[t], y4[t]} /. sol], {t, t0, tf}, 
 PlotLabels -> {"y1(t)", "y2(t)", "y3(t)", "y4(t)"}, 
 PlotLegends -> {"y1(t)", "y2(t)", "y3(t)", "y4(t)"}, 
 PlotLabel -> "Solutions to the System of ODEs", 
 AxesLabel -> {"t", "y(t)"}]

(*Define the system of ODEs*)
system = {y1'[t] == y2[t]^2, y2'[t] == -0.5*y1[t], y3'[t] == y4[t]^2, 
   y4'[t] == -0.5*y3[t]};

(*Initial conditions*)
initialConditions = {y1[0] == 1, y2[0] == 0, y3[0] == 0, y4[0] == 1};

(*Solve the system of ODEs and plot the solutions*)
sol = NDSolve[{system, initialConditions}, {y1, y2, y3, y4}, {t, t0, 
    tf}];
Plot[Evaluate[{y1[t], y2[t], y3[t], y4[t]} /. sol], {t, t0, tf}, 
 PlotLabels -> {"y1(t)", "y2(t)", "y3(t)", "y4(t)"}, 
 PlotLegends -> {"y1(t)", "y2(t)", "y3(t)", "y4(t)"}, 
 PlotLabel -> "Test Case 2: Solutions to the System of ODEs", 
 AxesLabel -> {"t", "y(t)"}]


(*Define the system of ODEs*)
system = {y1'[t] == y2[t], y2'[t] == -10.5*y1[t], y3'[t] == y4[t], 
   y4'[t] == -0.5*y3[t]};

(*Initial conditions*)
initialConditions = {y1[0] == 0, y2[0] == 1, y3[0] == 1, y4[0] == 0};

(*Solve the system of ODEs and plot the solutions*)
sol = NDSolve[{system, initialConditions}, {y1, y2, y3, y4}, {t, t0, 
    tf}];
Plot[Evaluate[{y1[t], y2[t], y3[t], y4[t]} /. sol], {t, t0, tf}, 
 PlotLabels -> {"y1(t)", "y2(t)", "y3(t)", "y4(t)"}, 
 PlotLegends -> {"y1(t)", "y2(t)", "y3(t)", "y4(t)"}, 
 PlotLabel -> "Test Case 3: Solutions to the System of ODEs", 
 AxesLabel -> {"t", "y(t)"}]