Math 2280-2 Example 5.6.4 with(LinearAlgebra): System matrix A:= Matrix(2,2,[4,2,3,-1]);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKSshKlE3

Initial condition x0 := Vector([7,3]);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpUz05Nw==

non-homogeneous term f := t -> -t*exp(-2*t)*Vector([15,4]);

Zio2I0kidEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqKDkkIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJEYrISIjRiwtSSdWZWN0b3JHRi82IzckIiM6IiIlRiwhIiJGJUYlRiU=

Take matrix exponential to get fundamental matrix solution expAt := t -> MatrixExponential(A*t);

Zio2I0kidEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLV9JLkxpbmVhckFsZ2VicmFHNiRJKF9zeXNsaWJHRiUlKnByb3RlY3RlZEdJMk1hdHJpeEV4cG9uZW50aWFsR0YlNiMqJkkiQUdGJSIiIjkkRjNGJUYlRiU=

expAt(t);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKCVHb0I=

Form integrand in formula (28) integrand := t->MatrixInverse(expAt(t)) . f(t);

Zio2I0kidEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLV9JLkxpbmVhckFsZ2VicmFHNiRGLUYsSS5NYXRyaXhJbnZlcnNlR0YlNiMtSSZleHBBdEdGJTYjOSQtSSJmR0YlRjdGJUYlRiU=

And the integral in formula 28 would be in maple (map means ''apply function to each element of matrix") map(int,integrand(s),s=0..t);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpKyVHMCI=

Actually do formula (28) sol:= t-> expAt(t) . ( x0 + map(int,integrand(s),s=0..t));

Zio2I0kidEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLUkmZXhwQXRHRiU2IzkkLCZJI3gwR0YlIiIiLUkkbWFwR0YsNiVJJGludEdGKy1JKmludGVncmFuZEdGJTYjSSJzR0YlL0Y9OyIiIUYyRjVGJUYlRiU=

simplify(sol(t));

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpb1hONw==

Reality check: does our solution satisfy DE? simplify(map(diff,sol(t),t) - A . sol(t) - f(t));

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpP2RvOQ==

TTdSMApJNVJUQUJMRV9TQVZFLzEyMzg5MDAwWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjJSIjIiMiIiUiIiQiIiMhIiJGJg==TTdSMApJNVJUQUJMRV9TQVZFLzEyMTQxODQwWColKWFueXRoaW5nRzYiNiJbZ2whIyUhISEiIyIjIiIoIiIkRiY=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