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(D^2-2*D-8);

Or I could solve the auxiliary equation.

>	solve({m^2-2*m-8=0},{m});

Thus two linearly independent solutions to the complementary equation (solutions to the corresponding homogeneous equation) are y[i] =

.

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

.

Assign this the name f.

>	f:=A*exp(x)+B*exp(-x);

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

>	diff(f,x,x)-2*diff(f,x)-8*f;

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

-9A = 2  and  -5B = -1.

Solve this system of equations.

>	solve({-9*A=2,-5*B=-1},{A,B});

The general solution would be

y = C[1]*e^(-2*x) + C[2]*e^(4*x) - (2/9)*e^x + (1/5)*e^(-x).

Below is Maple's solution to the ODE.

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

>	ode1:={diff(y(x),x,x)-2*diff(y(x),x)-8*y(x)=2*exp(x)-exp(-x)};

>	dsolve(ode1);

Here we put in some initial conditions.

>	ic:={y(0)=0,D(y)(0)=1};

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

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

>	ode2:=diff(y(x),x$2)-2*diff(y(x),x)-8*y(x)=2*exp(x)-exp(-x);

>	ic2:=y(0)=0,D(y)(0)=1;

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

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