Exercises

What is the solution of this ODE: $\frac{d y (t)}{d t} = λ y (t)$ exer1.gif ?

$y (t) = y_{0} \exp (λ t)$ exer2.gif

How are the solutions of these two ODEs different?
$\frac{d y (t)}{d t} = λ y (t)$ exer3.gif and $\frac{d f (x)}{d x} = β f (x)$ exer4.gif

The solutions are the SAME. The names of the unknown functions and variables have been changed but this does not fundamentally change the solution. The same common ODEs can be found in very different branches of engineering. If you know the solution in one case, then you also know it in the other cases. Your physical interpretation of the solution will be different, but the mathematics is the same.

If you can’t find the analytic function that is the solution of an ODE, then what else can you do?

Find a NUMERICAL solution to the ODE.

What makes an ODE first order?

The ODE involves terms that include relationships between the unknown function and its FIRST derivative.

PT0gUXVpeiB0aXRsZQo9PT0gRm9sbG93IHRoZXNlIGluc3RydWN0aW9ucyB3aGVuIHRha2luZyB0 aGlzIHF1aXouCj8/IFdoaWNoIGZvcm11bGEgZ2l2ZXMgdGhlIHZhbHVlIG9mIHRoZSBkZWZpbml0 ZSBpbnRlZ3JhbCA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhN TCI+CiA8c2VtYW50aWNzPgogIDxtcm93PgogICA8bXN0eWxlIGRpc3BsYXlzdHlsZT0idHJ1ZSI+ CiAgICA8bXJvdz48bXVuZGVyb3Zlcj4KICAgICA8bW8+woLDoMK0PC9tbz4KICAgICA8bW4+MDwv bW4+CiAgICAgPG1pPm48L21pPgogICAgPC9tdW5kZXJvdmVyPgogICAgPG1yb3c+CiAgICAgPG1p PmE8L21pPjxtaT54PC9taT4KICAgIDwvbXJvdz4KICAgPC9tcm93PgogICAKICA8L21zdHlsZT4K IDwvbXJvdz4KPGFubm90YXRpb24gZW5jb2Rpbmc9Ik1hdGhUeXBlLU1URUYiPgpNYXRoVHlwZUBN VEVGQDVANUArPWZlYWFmaWFydDFldjFhcWF0Q3ZBVWZlQlNqdXlaTDJ5ZDlnekxidnlOdjJDYWVy YnVMd0JMbmhpb3YyREdpMUJUZk1CYWVYYXRMeEJJOWdCYWVyYmQ5d0RZTHd6WWJJdExEaGFycXF0 dWJzcjRyTkNIYkdlYUdxaVZ1MEplOXNxcXJwZXBDMHhiYkw4RjRycXFyRmZwZWVhMHhlOUxxPUpj OXZxYXFwZXBtMHhiYmE5cHdlOVE4ZnMwPXlxYXFwZXBhZTlwZzBGaXJwZXBlS2tGcjB4ZnI9eGZy PXhiOWFkYmFxYWFlR2FjaUdhYWlhYWJlcWFhbWFhYmFhYmFhR2NiYVdhYThxQ2FlYWFjYVdHSGJH YWFtaUVhYVdjYmFHYWFHaW1hYXFhYWlhYWQ2Z2FhMEdhZXk0a0lpcGFhYWFAM0JFN0A8L2Fubm90 YXRpb24+Cjwvc2VtYW50aWNzPgo8L21hdGg+W1tBbHRNYXRoTUxJbWFnZTplcW4wMDcuZ2lmfG1h dGggaW1hZ2VdXT8KKysgPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9N YXRoTUwiPgogPHNlbWFudGljcz4KICA8bXJvdz4KICAgPG1mcmFjPgogICAgPG1uPjE8L21uPgog ICAgPG1uPjI8L21uPgogICA8L21mcmFjPgogICA8bWk+YTwvbWk+PG1zdXA+CiAgICA8bWk+bjwv bWk+CiAgICA8bW4+MjwvbW4+CiAgIDwvbXN1cD4KICAgCiAgPC9tcm93PgogPGFubm90YXRpb24g ZW5jb2Rpbmc9Ik1hdGhUeXBlLU1URUYiPgogTWF0aFR5cGVATVRFRkA1QDVAKz1mZWFhZmlhcnQx ZXYxYXFhdEN2QVVmZUJTanV5WkwyeWQ5Z3pMYnZ5TnYyQ2FlcmJ1THdCTG5oaW92MkRHaTFCVGZN QmFlWGF0THhCSTlnQmFlcmJkOXdEWUx3elliSXRMRGhhcnFxdHVic3I0ck5DSGJHZWFHcWlWdTBK ZTlzcXFycGVwQzB4YmJMOEY0cnFxckZmcGVlYTB4ZTlMcT1KYzl2cWFxcGVwbTB4YmJhOXB3ZTlR OGZzMD15cWFxcGVwYWU5cGcwRmlycGVwZUtrRnIweGZyPXhmcj14YjlhZGJhcWFhZUdhY2lHYWFp YWFiZXFhYW1hYWJhYWJhYUdjYmFXYWFTYWFhZWFhY2FhSVhhYWFiYUdhYUdPbWFhYWFjYVdHSGJH YWFtT0JhbWFhQ2FhYWxlcWFiYUdhYUdPbWFhYWFhYWFAM0EzN0A8L2Fubm90YXRpb24+CiA8L3Nl bWFudGljcz4KPC9tYXRoPltbQWx0TWF0aE1MSW1hZ2U6ZXFuMDA4LmdpZl1dCistIDxtYXRoIHht bG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj4KIDxzZW1hbnRpY3M+CiAg PG1yb3c+CiAgIDxtZnJhYz4KICAgIDxtbj4xPC9tbj4KICAgIDxtbj4yPC9tbj4KICAgPC9tZnJh Yz4KICAgPG1zdXA+CiAgICA8bWk+YTwvbWk+CiAgICA8bW4+MjwvbW4+CiAgIDwvbXN1cD4KICAg PG1zdXA+CiAgICA8bWk+bjwvbWk+CiAgICA8bW4+MjwvbW4+CiAgIDwvbXN1cD4KICAgCiAgPC9t cm93PgogPGFubm90YXRpb24gZW5jb2Rpbmc9Ik1hdGhUeXBlLU1URUYiPgogTWF0aFR5cGVATVRF RkA1QDVAKz1mZWFhZmlhcnQxZXYxYXFhdEN2QVVmZUJTanV5WkwyeWQ5Z3pMYnZ5TnYyQ2FlcmJ1 THdCTG5oaW92MkRHaTFCVGZNQmFlWGF0THhCSTlnQmFlcmJkOXdEWUx3elliSXRMRGhhcnFxdHVi c3I0ck5DSGJHZWFHcWlWdTBKZTlzcXFycGVwQzB4YmJMOEY0cnFxckZmcGVlYTB4ZTlMcT1KYzl2 cWFxcGVwbTB4YmJhOXB3ZTlROGZzMD15cWFxcGVwYWU5cGcwRmlycGVwZUtrRnIweGZyPXhmcj14 YjlhZGJhcWFhZUdhY2lHYWFpYWFiZXFhYW1hYWJhYWJhYUdjYmFXYWFTYWFhZWFhY2FhSVhhYWFi YUdhYUdPbWFhYWFjYVdHSGJXYWFXYmFhU3FhYmVhYWNhYUlZYWFhYU9HYWFtT0JhbWFhQ2FhYWxl cWFiYUdhYUdPbWFhYWFhYWFAM0IyQUA8L2Fubm90YXRpb24+CiA8L3NlbWFudGljcz4KPC9tYXRo PltbQWx0TWF0aE1MSW1hZ2U6ZXFuMDA5LmdpZl1d