diff --git a/content/differentialequations.md b/content/differentialequations.md index d68ae41ff1ae67800865d8f43143fdec10c2206e..a47af0ee9e6c237fec0d26a3a2930ecf06c12a4a 100644 --- a/content/differentialequations.md +++ b/content/differentialequations.md @@ -131,7 +131,7 @@ $$ x(t) = A e^{\lambda_{+}t} + B e^{\lambda_{-}t}, $$ (secondorderodesol2) -where $A$ and $B$ are set by either initial or boundary conditions. Since the $\lambda_\pm$ may be complex, so may $A$ and $B$; it's their combination that should give a real number (as $x(t)$ is real), see problem {numref}`app:solvingde`.\ref{prob:rewritesolutionsecondorderode}. +where $A$ and $B$ are set by either initial or boundary conditions. Since the $\lambda_\pm$ may be complex, so may $A$ and $B$; it's their combination that should give a real number (as $x(t)$ is real), see {numref}`pb:secondorderodesolutions`a. In the case that equation {eq}`secondorderodesol1` gives only one solution, the corresponding exponential function is still a solution of equation {eq}`secondorderodeconstcoeff`, but it is not the most general one, as we only can put a single undetermined constant in front of it. We therefore need a second, independent solution. To guess one, here's a third useful trick<sup>[^4]</sup>: take the derivative of our known solution, $e^{\lambda t}$, with respect to the parameter $\lambda$. This gives a second Ansatz: $t e^{\lambda t}$, where $\lambda = -b/2a$. Substituting this Ansatz into equation {eq}`secondorderodeconstcoeff` for the case that $c = b^2/2a$, we find: diff --git a/content/linearalgebra.md b/content/linearalgebra.md index 20c20e5242ff75a89a7a84cfd3f0dfc11e88fff7..2f2f1989728f95ad8594325df47517deb55476ab 100644 --- a/content/linearalgebra.md +++ b/content/linearalgebra.md @@ -144,7 +144,7 @@ A = \begin{pmatrix} 5 \\ 4 \\ 3 \end{pmatrix}. ``` -Find $\bm{y}$. +Find $\bm{y}$. --- **Solution** @@ -262,6 +262,7 @@ A = \begin{pmatrix} 2 & 1 \\ 2 & 4 \end{pmatrix}. ``` + --- **Solution** We write the matrix $A$ in combination with the identity matrix, then apply row-reduction until we've reduced $A$ itself to the identity matrix. @@ -397,6 +398,7 @@ Find the eigenvalues and eigenvectors of the following matrices: ```{math} A = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}, \qquad B = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 0 & 1 \end{pmatrix}. ``` + --- **Solution** For the matrix $A$, we can write down and solve the characteristic equation with ease: