Second-Order Linear Homogeneous
Differential Equations with
Constant Coefficients

Part 1. The Equation y'' + by = 0 with b > 0

Enter the coefficients for the differential equation and the differential equation.

a  =  0;
b  =  1;
DE  =  y''[t]  +  a*y'[t] + b*y[t] == 0;

Plot the solution of the initial value problem.

solution  =  
        NDSolve[{DE,  y[0] == 1,  y'[0] == 0},  
          y,  {t, 0, 15}];
Plot[Evaluate[y[t] /. solution], {t, 0, 15}, 
        PlotStyle -> RGBColor[0, 0, 1], 
        AxesLabel -> {"t", "y[t]"}];

Change the value of b successively, and replot for each value.

Answer to step 1:

Reset b = 1, change y0 to 2, then to -1, and replot for each value.

Answer to step 2:

Reset y0 = 1, change y1 to 2, then to 3, and replot for each value.

Answer to step 3:

Change y1 to-2, then to-3, and replot for each value.

Answer to step 4:

Part 2. The Equation y'' + b y = 0 with b < 0

Enter the coefficients for the differential equation and the differential equation.

a  =  0;
b  =  -1;
y0  =  1;
y1  =  0;
DE  =  y''[t] + a*y'[t] + b*y[t] == 0;

Plot the solution of the initial value problem.

solution  =  
        NDSolve[{DE,  y[0] == 1,  y'[0] == 0},  
          y,  {t, 0, 4}];
Plot[Evaluate[y[t] /. solution], {t, 0, 4}, 
        PlotStyle -> RGBColor[0, 0, 1], 
        AxesLabel -> {"t", "y[t]"}];

Change the value of b successively, and replot for each value.

Answer to step 1:

Reset b = -1, and leave y0=1. Change y1 to -1, then to -1.1, then to -0.9, and replot for each value.

Adjust t and y scales as necessary to see what's going on.

Answer to step 2:

Part 3. The Equation y'' + a*y' + b*y = 0 for a ≠ 0.

Enter the coefficients for the differential equation and the differential equation.

a  =  0.5;
b  =  1;
DE  =  y''[t] + a*y'[t] + b*y[t] == 0;

Plot the solution of the initial value problem.

solution  =  
        NDSolve[{DE,  y[0] == 1,  y'[0] == 0},  
          y,  {t, 0, 10}];
Plot[Evaluate[y[t] /. solution], {t, 0, 10}, 
        PlotStyle -> RGBColor[0, 0, 1], 
        AxesLabel -> {"t", "y[t]"}];

Change the value of a successively, and replot for each value.

Answer to step 1:

Change the value of a through successive negative values, and replot for each value.

Answer to step 2:

For a = -1, plot the solution on successively longer t intervals.

Answer to step 3:

Answer to step 4:

Part 4. Summary

Answer the summary questions here.