A DE Example

Second order, nonhomogeneous, constant coefficient

Here are some maple commands to assist you in solving a second order, constant coefficient, nonhomogeneous differential equation.  The equation being solved is

First I will factor the operator.

>	factor(2*D^2+7*D+6);

Or I could solve the auxiliary equation.

>	solve({2*m^2+7*m+6=0},{m});

Thus the general solution to the corresponding homogeneous equation would be

.

The form for a particular solution would be y[p] =

.

Assign this the name f.

>	f:=A*sin(t)+B*cos(t);

Substitute f in for y in the left side of the original differentail equation.

>	2*diff(f,t,t)+7*diff(f,t)+6*f;

Set the expression above equal to the right side of the differential equation and equate coefficients of like terms to get

4A - 7B = 2  and  7A + 4B = 0.

Solve this system of equations.

>	solve({4*A-7*B=2,7*A+4*B=0},{A,B});

The general solution would be

.

Below is Maple's solution to the ODE.

Can you see that it agrees with the solution given above?

>	ode1:={2*diff(y(t),t,t)+7*diff(y(t),t)+6*y(t)=2*sin(t)};

>	dsolve(ode1);

Here we put in some initial conditions.

>	ic:={y(0)=3,D(y)(0)=0};

>	dsolve(ode1 union ic,y(t));

Here is some alternative syntax for solving the same differential equation along with initial conditions.

>	ode2:=2*diff(y(t),t$2)+7*diff(y(t),t)+6*y(t)=2*sin(t);

>	ic2:=y(0)=3,D(y)(0)=0;

>	dsolve({ode2,ic2},{y(t)});

Here are the graphs of the transient solution (in green), steady state solution (in blue), and the solution (in red).

>	with(plots):

>	Transient:=plot((-47/5)*exp(-2*t)+(164/13)*exp(-3*t/2),t=0..4*Pi,y=-2..4,thickness=3,color=green):

>	SteadyState:=plot((8/65)*sin(t)-(14/65)*cos(t),t=0..4*Pi,y=-2..4,thickness=3,color=blue):

>	Solution:=plot((-47/5)*exp(-2*t)+(164/13)*exp(-3*t/2)+(8/65)*sin(t)-(14/65)*cos(t),t=0..4*Pi,y=-2..4,thickness=3,color=red):

>	display(Transient,SteadyState,Solution);

>