As this is my first time dealing with ODE's, I'm having alot of trouble proving some of the theorems that are given to me involving homogeneous and non homogeneous systems. For example
Theorem
Let $\phi_{1}(t),…,\phi_{n-1}(t)$ be solutions of the homogeneous
system $\dot x = A(t)x$ and let $x(t)$ be a solution of the non
homogeneous system $\dot x = A(t)x + b(t)$ where $A(t),b(t)$ are
continuous. Then $Z=\det(\phi_{1}(t),…,\phi_{n-1}(t), x(t))$
satisfies the non homogeneous Abel's formula $$\dot Z =
\text{tr}(A(t))Z+\det(\phi_{1}(t),\dots,\phi_{n-1}(t),b(t))$$
Probably my problem is with linear algebra more than ODE's but still, I haven't been able to prove it nor find a proof online. Any help would be greatly appreciated. Thanks so much in advance.
Best Answer
In general, suppose $B$ and $Y=[y_1,\ldots,y_n]$ are two square matrices of the same sizes. Let $\{e_1,\ldots,e_n\}$ be the standard basis. When $Y$ is nonsingular, we have \begin{aligned} &\det\left(Y^{-1}BYe_1,e_2,\ldots,e_n\right) +\det\left(e_1,Y^{-1}BYe_2,\ldots,e_n\right) +\cdots +\det\left(e_1,\ldots,e_{n-1},Y^{-1}BYe_n\right)\\ &=\operatorname{tr}(Y^{-1}BY)=\operatorname{tr}(B). \end{aligned} Hence $$ \det\left(By_1,y_2,\ldots,y_n\right) +\det\left(y_1,By_2,\ldots,y_n\right) +\cdots +\det\left(y_1,\ldots,y_{n-1},By_n\right)=\operatorname{tr}(B)\det(Y)\tag{1} $$ and by a continuity argument, $(1)$ is true for singular $Y$ as well.
Now, return to your question. By the product rule, \begin{aligned} \dot{Z} &= \det\left(\dot{\phi}_1,\phi_2,\ldots,\phi_{n-1},x\right) +\det\left(\phi_1,\dot{\phi}_2,\ldots,\phi_{n-1},x\right) +\cdots +\det\left(\phi_1,\phi_2,\ldots,\phi_{n-1},\dot{x}\right)\\ &= \det\left(A\phi_1,\phi_2,\ldots,\phi_{n-1},x\right) +\det\left(\phi_1,A\phi_2,\ldots,\phi_{n-1},x\right) +\cdots +\det\left(\phi_1,\phi_2,\ldots,\phi_{n-1},Ax+b\right)\\ &= \det\left(A\phi_1,\phi_2,\ldots,\phi_{n-1},x\right) +\det\left(\phi_1,A\phi_2,\ldots,\phi_{n-1},x\right) +\cdots +\det\left(\phi_1,\phi_2,\ldots,\phi_{n-1},Ax\right)\\ &\phantom{=}+\det\left(\phi_1,\phi_2,\ldots,\phi_{n-1},b\right), \end{aligned} Put $B=A$ and $[y_1,\ldots,y_n]=[\phi_1,\ldots,\phi_{n-1},x]$ into $(1)$, the result follows.