Solving Two Dimensional Systems of Linear Differential Equations. restart; with(LinearAlgebra): A utiity procedure. buildsys := proc(mat) {diff(x(t),t)=mat[1,1]*x(t)+mat[1,2]*y(t), diff(y(t),t)=mat[2,1]*x(t)+mat[2,2]*y(t)}; end;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlidWlsZHN5c0dGKGYqNiNJJG1hdEdGKEYoRihGKDwkLy1JJWRpZmZHRiY2JC1JInhHRig2I0kidEdGKEY5LCYqJiY5JDYkIiIiRj9GP0Y2Rj9GPyomJkY9NiRGPyIiI0Y/LUkieUdGKEY4Rj9GPy8tRjQ2JEZERjksJiomJkY9NiRGQ0Y/Rj9GNkY/Rj8qJiZGPTYkRkNGQ0Y/RkRGP0Y/RihGKEYoNyNGLg==

Diagonializable, real eigenvalues. A := Matrix([[-4, 6], [-3, 5]]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJBR0YoLUknUlRBQkxFR0YoNiUiKClHKGUoLUknTUFUUklYR0YoNiM3JDckISIlIiInNyQhIiQiIiZJJ01hdHJpeEdGJTcjLUY8NiMvSSQlaWRHRihGMQ==

sys := buildsys(A);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzeXNHRig8JC8tSSVkaWZmR0YmNiQtSSJ5R0YoNiNJInRHRihGNiwmLUkieEdGKEY1ISIkRjMiIiYvLUYxNiRGOEY2LCZGOCEiJUYzIiInNyNGLg==

Initial Condition xno := <a,b>;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR4bm9HRigtSSdSVEFCTEVHRig2JSIoR1JfKC1JJ01BVFJJWEdGKDYjNyQ3I0kiYUdGKDcjSSJiR0YoJkknVmVjdG9yR0YlNiNJJ2NvbHVtbkdGKDcjLUY6NiMvSSQlaWRHRihGMQ==

Maple's Solution sol := dsolve(sys union {x(0)=a, y(0)=b}, [x(t),y(t)]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzb2xHRig8JC8tSSJ5R0YoNiNJInRHRigsJiomLCZJImFHRighIiJJImJHRigiIiMiIiItSSRleHBHRiU2IywkRjNGOkY7RjsqJiwmRjdGO0Y5RjhGOy1GPTYjLCRGM0Y4RjtGOy8tSSJ4R0YoRjIsJkY1RjtGQEY6NyNGLg==

Now, we'll solve it. evv := Eigenvectors(A);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRldnZHRig2JC1JJ1JUQUJMRUdGKDYlIilvbjQ4LUknTUFUUklYR0YoNiM3JDcjIiIjNyMhIiImSSdWZWN0b3JHRiU2I0knY29sdW1uR0YoLUYwNiUiKGNDOSktRjQ2IzckNyQiIiJGODckRkZGRkknTWF0cml4R0YlNyQtRjs2Iy9JJCVpZEdGKEYyLUZINiMvRk1GQQ==

P:=evv[2];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJQR0YoLUknUlRBQkxFR0YoNiUiKGNDOSktSSdNQVRSSVhHRig2IzckNyQiIiIiIiM3JEY3RjdJJ01hdHJpeEdGJTcjLUY6NiMvSSQlaWRHRihGMQ==

P^(-1).A.P;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSIoWz87KS1JJ01BVFJJWEdGKDYjNyQ3JCIiIyIiITckRjYhIiJJJ01hdHJpeEdGJTcjLUY5NiMvSSQlaWRHRihGLw==

zmat := DiagonalMatrix([exp(2*t), exp(-t)]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSV6bWF0R0YoLUknUlRBQkxFR0YoNiUiKE8oKkgoLUknTUFUUklYR0YoNiM3JDckLUkkZXhwR0YlNiMsJEkidEdGKCIiIyIiITckRj0tRjg2IywkRjshIiJJJ01hdHJpeEdGJTcjLUZDNiMvSSQlaWRHRihGMQ==

xsol := P.zmat.P^(-1).xno;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSV4c29sR0YoLUknUlRBQkxFR0YoNiUiKTMjKTQ4LUknTUFUUklYR0YoNiM3JDcjLCYqJiwmLUkkZXhwR0YlNiMsJEkidEdGKCIiIyEiIi1GOzYjLCRGPkZARj8iIiJJImFHRihGREZEKiYsJkY6Rj9GQSEiI0ZESSJiR0YoRkRGRDcjLCYqJiwmRjpGQEZBRkRGREZFRkRGRComLCZGOkY/RkFGQEZERklGREZEJkknVmVjdG9yR0YlNiNJJ2NvbHVtbkdGKDcjLUZQNiMvSSQlaWRHRihGMQ==

Rearrange to compare with Maple. map(collect, %, [exp(2*t), exp(-t)]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSIpQ3FNOC1JJ01BVFJJWEdGKDYjNyQ3IywmKiYsJkkiYUdGKCEiIkkiYkdGKCIiIyIiIi1JJGV4cEdGJTYjLCRJInRHRihGO0Y8RjwqJiwmRjhGO0Y6ISIjRjwtRj42IywkRkFGOUY8Rjw3IywmRjZGPComLCZGOEY8RjpGOUY8RkVGPEY8JkknVmVjdG9yR0YlNiNJJ2NvbHVtbkdGKDcjLUZMNiMvSSQlaWRHRihGLw==

Diagonalizable, nonreal eigenvalues. A := Matrix([[11, -6], [15, -7]]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJBR0YoLUknUlRBQkxFR0YoNiUiKTN3XDktSSdNQVRSSVhHRig2IzckNyQiIzYhIic3JCIjOiEiKEknTWF0cml4R0YlNyMtRjw2Iy9JJCVpZEdGKEYx

sys := buildsys(A);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzeXNHRig8JC8tSSVkaWZmR0YmNiQtSSJ4R0YoNiNJInRHRihGNiwmRjMiIzYtSSJ5R0YoRjUhIicvLUYxNiRGOUY2LCZGMyIjOkY5ISIoNyNGLg==

dsol:=dsolve(sys union {x(0)=a, y(0)=b}, [x(t), y(t)]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkc29sR0YoPCQvLUkieUdGKDYjSSJ0R0YoKiYtSSRleHBHRiU2IywkRjMiIiMiIiIsJiomLCZJImJHRighIiRJImFHRigiIiZGOi1JJHNpbkdGJTYjLCRGMyIiJEY6RjoqJkY+RjotSSRjb3NHRiVGREY6RjpGOi8tSSJ4R0YoRjIsJComRjVGOiwqRjxGRiomRj1GOkZIRjpGOkZHRkYqJkY+RjpGQkY6ISIiRjojRjpGQTcjRi4=

evv := Eigenvectors(A);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRldnZHRig2JC1JJ1JUQUJMRUdGKDYlIiklKVJJOC1JJ01BVFJJWEdGKDYjNyQ3I14kIiIjIiIkNyNeJEY5ISIkJkknVmVjdG9yR0YlNiNJJ2NvbHVtbkdGKC1GMDYlIig/OisqLUY0NiM3JDckXiQjRjoiIiYjIiIiRkteJEZKIyEiIkZLNyRGTUZNSSdNYXRyaXhHRiU3JC1GPjYjL0kkJWlkR0YoRjItRlI2Iy9GV0ZE

P := evv[2];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJQR0YoLUknUlRBQkxFR0YoNiUiKD86KyotSSdNQVRSSVhHRig2IzckNyReJCMiIiQiIiYjIiIiRjpeJEY4IyEiIkY6NyRGPEY8SSdNYXRyaXhHRiU3Iy1GQTYjL0kkJWlkR0YoRjE=

p1 := Column(P,1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNwMUdGKC1JJ1JUQUJMRUdGKDYlIilrXUk4LUknTUFUUklYR0YoNiM3JDcjXiQjIiIkIiImIyIiIkY6NyNGPCZJJ1ZlY3RvckdGJTYjSSdjb2x1bW5HRig3Iy1GPjYjL0kkJWlkR0YoRjE=

u := map(Re,p1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ1R0YoLUknUlRBQkxFR0YoNiUiKSlbL0wiLUknTUFUUklYR0YoNiM3JDcjIyIiJCIiJjcjIiIiJkknVmVjdG9yR0YlNiNJJ2NvbHVtbkdGKDcjLUY8NiMvSSQlaWRHRihGMQ==

v := -map(Im, p1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ2R0YoLUknUlRBQkxFR0YoNiUiKTtZaDktSSdNQVRSSVhHRig2IzckNyMjISIiIiImNyMiIiEmSSdWZWN0b3JHRiU2I0knY29sdW1uR0YoNyMtRjw2Iy9JJCVpZEdGKEYx

u - I*v;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSIpc1VJOC1JJ01BVFJJWEdGKDYjNyQ3I14kIyIiJCIiJiMiIiJGODcjRjomSSdWZWN0b3JHRiU2I0knY29sdW1uR0YoNyMtRjw2Iy9JJCVpZEdGKEYv

Q := <u|v>;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJRR0YoLUknUlRBQkxFR0YoNiUiKU9oKVIiLUknTUFUUklYR0YoNiM3JDckIyIiJCIiJiMhIiJGOTckIiIiIiIhSSdNYXRyaXhHRiU3Iy1GPzYjL0kkJWlkR0YoRjE=

B:=Q^(-1).A.Q;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJCR0YoLUknUlRBQkxFR0YoNiUiKVM/MjktSSdNQVRSSVhHRig2IzckNyQiIiMhIiQ3JCIiJEY3SSdNYXRyaXhHRiU3Iy1GOzYjL0kkJWlkR0YoRjE=

zmat := << exp(2*t)*cos(3*t)|-exp(2*t)*sin(3*t)>, <exp(2*t)*sin(3*t)| exp(2*t)*cos(3*t)>>;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSV6bWF0R0YoLUknUlRBQkxFR0YoNiUiKCkzZCcpLUknTUFUUklYR0YoNiM3JDckKiYtSSRleHBHRiU2IywkSSJ0R0YoIiIjIiIiLUkkY29zR0YlNiMsJEY8IiIkRj4sJComRjhGPi1JJHNpbkdGJUZBRj4hIiI3JEZFRjdJJ01hdHJpeEdGJTcjLUZKNiMvSSQlaWRHRihGMQ==

xsol := Q.zmat.Q^(-1).xno;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSV4c29sR0YoLUknUlRBQkxFR0YoNiUiKEtDbSktSSdNQVRSSVhHRig2IzckNyMsJiomLCYqJi1JJGV4cEdGJTYjLCRJInRHRigiIiMiIiItSSRzaW5HRiU2IywkRj8iIiRGQUZGKiZGO0ZBLUkkY29zR0YlRkRGQUZBRkFJImFHRihGQUZBKihGO0ZBRkJGQUkiYkdGKEZBISIjNyMsJiooRjtGQUZCRkFGSkZBIiImKiYsJkZHRkFGOiEiJEZBRkxGQUZBJkknVmVjdG9yR0YlNiNJJ2NvbHVtbkdGKDcjLUZVNiMvSSQlaWRHRihGMQ==

To check that this is the same as Maple's solution, create differences {x(t)-xsol[1], y(t)-xsol[2]};

PCQsKC1JInhHNiI2I0kidEdGJiIiIiomLCYqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLCRGKCIiI0YpLUkkc2luR0YvNiMsJEYoIiIkRilGOSomRi1GKS1JJGNvc0dGL0Y3RilGKUYpSSJhR0YmRikhIiIqKEYtRilGNUYpSSJiR0YmRilGNCwoLUkieUdGJkYnRikqKEYtRilGNUYpRj1GKSEiJiomLCZGOkYpRiwhIiRGKUZARilGPg==

Now substitute Maple's solutions for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSIuRidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURj0= subs(dsol, %);

PCQsKComLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJEkidEdGKiIiIyIiIiwqKiYsJkkiYkdGKiEiJEkiYUdGKiIiJkYvLUkkc2luR0YnNiMsJEYtIiIkRi9GOyomRjJGLy1JJGNvc0dGJ0Y5Ri9GLyomRjNGL0Y9Ri9GOyomRjNGL0Y3Ri8hIiJGLyNGL0Y2KiYsJiomRiVGL0Y3Ri9GOyomRiVGL0Y9Ri9GL0YvRjVGL0ZBKihGJUYvRjdGL0YzRi9GLiwoKiZGJUYvLCZGMUYvRj9GL0YvRi8qKEYlRi9GN0YvRjVGLyEiJiomLCZGRkYvRkVGNEYvRjNGL0ZB

map(simplify, %);

PCMiIiE=

Nondiagonalizable. A := Matrix([[-8/3, 4/3], [-1/3, -4/3]]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJBR0YoLUknUlRBQkxFR0YoNiUiKClHMGEtSSdNQVRSSVhHRig2IzckNyQjISIpIiIkIyIiJUY5NyQjISIiRjkjISIlRjlJJ01hdHJpeEdGJTcjLUZBNiMvSSQlaWRHRihGMQ==

sys := buildsys(A);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzeXNHRig8JC8tSSVkaWZmR0YmNiQtSSJ5R0YoNiNJInRHRihGNiwmLUkieEdGKEY1IyEiIiIiJEYzIyEiJUY8Ly1GMTYkRjhGNiwmRjgjISIpRjxGMyMiIiVGPDcjRi4=

dsol := dsolve(sys union {y(0)=b, x(0)=a}, [x(t), y(t)]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkc29sR0YoPCQvLUkieUdGKDYjSSJ0R0YoKiYtSSRleHBHRiU2IywkRjMhIiMiIiIsJkkiYkdGKEY6KiYsJkkiYUdGKCMhIiIiIiRGPCMiIiNGQkY6RjNGOkY6RjovLUkieEdGKEYyKiZGNUY6LCZGPUZERj9GOkY6NyNGLg==

evv := Eigenvectors(A);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRldnZHRig2JC1JJ1JUQUJMRUdGKDYlIilbZzU2LUknTUFUUklYR0YoNiM3JDcjISIjRjcmSSdWZWN0b3JHRiU2I0knY29sdW1uR0YoLUYwNiUiKGtzKCoqLUY0NiM3JDckIiIjIiIhNyQiIiJGRUknTWF0cml4R0YlNyQtRjk2Iy9JJCVpZEdGKEYyLUZINiMvRk1GPw==

u:= <2, 1>;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ1R0YoLUknUlRBQkxFR0YoNiUiKW8+NDgtSSdNQVRSSVhHRig2IzckNyMiIiM3IyIiIiZJJ1ZlY3RvckdGJTYjSSdjb2x1bW5HRig3Iy1GOjYjL0kkJWlkR0YoRjE=

Pick some vector independent of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5 v := < -1,3>;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ2R0YoLUknUlRBQkxFR0YoNiUiKStHVDgtSSdNQVRSSVhHRig2IzckNyMhIiI3IyIiJCZJJ1ZlY3RvckdGJTYjSSdjb2x1bW5HRig3Iy1GOjYjL0kkJWlkR0YoRjE=

A.v;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSIpO1dJOC1JJ01BVFJJWEdGKDYjNyQ3IyMiIz8iIiQ3IyMhIzZGNyZJJ1ZlY3RvckdGJTYjSSdjb2x1bW5HRig3Iy1GOzYjL0kkJWlkR0YoRi8=

A.v - (-2)*v;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSIpR2ZyNy1JJ01BVFJJWEdGKDYjNyQ3IyMiIzkiIiQ3IyMiIihGNyZJJ1ZlY3RvckdGJTYjSSdjb2x1bW5HRig3Iy1GOzYjL0kkJWlkR0YoRi8=

This is 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 Adjust LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5 v := v/(7/3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJ2R0YoLUknUlRBQkxFR0YoNiUiKSNmVkYiLUknTUFUUklYR0YoNiM3JDcjIyEiJCIiKDcjIyIiKkY5JkknVmVjdG9yR0YlNiNJJ2NvbHVtbkdGKDcjLUY9NiMvSSQlaWRHRihGMQ==

A.v - (-2)*v;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSIpJyozIVwiLUknTUFUUklYR0YoNiM3JDcjIiIjNyMiIiImSSdWZWN0b3JHRiU2I0knY29sdW1uR0YoNyMtRjg2Iy9JJCVpZEdGKEYv

This is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5 Tis normalization gives us the 1 in the matrix B below. P := <u|v>;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJQR0YoLUknUlRBQkxFR0YoNiUiKWMhekwiLUknTUFUUklYR0YoNiM3JDckIiIjIyEiJCIiKDckIiIiIyIiKkY6SSdNYXRyaXhHRiU3Iy1GPzYjL0kkJWlkR0YoRjE=

B := P^(-1).A.P;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJCR0YoLUknUlRBQkxFR0YoNiUiKVM/TjUtSSdNQVRSSVhHRig2IzckNyQhIiMiIiI3JCIiIUY3SSdNYXRyaXhHRiU3Iy1GOzYjL0kkJWlkR0YoRjE=

zmat := << exp(-2*t) | t*exp(-2*t)>, <0| exp(-2*t)>>;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSV6bWF0R0YoLUknUlRBQkxFR0YoNiUiKSV5aUkiLUknTUFUUklYR0YoNiM3JDckLUkkZXhwR0YlNiMsJEkidEdGKCEiIyomRjsiIiJGN0Y+NyQiIiFGN0knTWF0cml4R0YlNyMtRkE2Iy9JJCVpZEdGKEYx

xsol := P.zmat.P^(-1).xno;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSV4c29sR0YoLUknUlRBQkxFR0YoNiUiKU9GWDgtSSdNQVRSSVhHRig2IzckNyMsJiomLCYtSSRleHBHRiU2IywkSSJ0R0YoISIjIiIiKiZGPkZARjpGQCNGPyIiJEZASSJhR0YoRkBGQCooRj5GQEY6RkBJImJHRihGQCMiIiVGQzcjLCYqKEY+RkBGOkZARkRGQCMhIiJGQyomLCZGOkZARkEjIiIjRkNGQEZGRkBGQCZJJ1ZlY3RvckdGJTYjSSdjb2x1bW5HRig3Iy1GUjYjL0kkJWlkR0YoRjE=

{x(t)-xsol[1], y(t)-xsol[2]};

PCQsKC1JInlHNiI2I0kidEdGJiIiIiooRihGKS1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLCRGKCEiI0YpSSJhR0YmRikjRikiIiQqJiwmRitGKSomRihGKUYrRikjIiIjRjVGKUkiYkdGJkYpISIiLCgtSSJ4R0YmRidGKSomLCZGK0YpRjgjRjJGNUYpRjNGKUY8KihGKEYpRitGKUY7RikjISIlRjU=

subs(dsol, %);

PCQsKComLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJEkidEdGKiEiIyIiIiwmSSJiR0YqRi8qJiwmSSJhR0YqIyEiIiIiJEYxIyIiI0Y3Ri9GLUYvRi9GL0YvKihGLUYvRiVGL0Y0Ri8jRi9GNyomLCZGJUYvKiZGLUYvRiVGL0Y4Ri9GMUYvRjYsKComRiVGLywmRjJGOUY0Ri9GL0YvKiYsJkYlRi9GPiNGLkY3Ri9GNEYvRjYqKEYtRi9GJUYvRjFGLyMhIiVGNw==

map(simplify, %);

PCMiIiE=

TTdSMApJNVJUQUJMRV9TQVZFLzEzMDk2NzY4WColKmFsZ2VicmFpY0c2IjYiW2dsISMlISEhIiMiIyIiIyEiIkYmTTdSMApJNFJUQUJMRV9TQVZFLzgxNDI0NTZYLCUqYWxnZWJyYWljRzYiNiJbZ2whIiUhISEjJSIjIiMiIiJGJyIiI0YnRiY=TTdSMApJNFJUQUJMRV9TQVZFLzgxNDI0NTZYLCUqYWxnZWJyYWljRzYiNiJbZ2whIiUhISEjJSIjIiMiIiJGJyIiI0YnRiY=TTdSMApJNVJUQUJMRV9TQVZFLzEzMzAzOTg0WColKmFsZ2VicmFpY0c2IjYiW2dsISMlISEhIiMiI14kIiIjIiIkXiRGKCEiJEYmTTdSMApJNFJUQUJMRV9TQVZFLzkwMDE1MjBYLCUqYWxnZWJyYWljRzYiNiJbZ2whIiUhISEjJSIjIiNeJCMiIiQiIiYjIiIiRipGLF4kRigjISIiRipGLEYmTTdSMApJNFJUQUJMRV9TQVZFLzkwMDE1MjBYLCUqYWxnZWJyYWljRzYiNiJbZ2whIiUhISEjJSIjIiNeJCMiIiQiIiYjIiIiRipGLF4kRigjISIiRipGLEYmTTdSMApJNVJUQUJMRV9TQVZFLzEzMzA1MDY0WColKmFsZ2VicmFpY0c2IjYiW2dsISMlISEhIiMiI14kIyIiJCIiJiMiIiJGKkYsRiY=TTdSMApJNVJUQUJMRV9TQVZFLzExMTA2MDQ4WColKmFsZ2VicmFpY0c2IjYiW2dsISMlISEhIiMiIyEiI0YnRiY=TTdSMApJNFJUQUJMRV9TQVZFLzk5NzcyNjRYLCUqYWxnZWJyYWljRzYiNiJbZ2whIiUhISEjJSIjIiMiIiMiIiIiIiFGKUYm