System of Diff Eq's

Note that $\dot{A} + \dot{B} \; = \; -(A+B) \hspace{2cm} \dot{A} - \dot{B} \; = \; A - B$ so that $A + B \; = \; \tfrac94e^{-t} \hspace{2cm} A - B \;= \; \tfrac14e^t$ Thus $A^2 - B^2 \; = \; \tfrac{9}{16}$ while $A = \tfrac18\big(9e^{-t} + e^t\big) > 0$ , so that $A \; = \; \sqrt{B^2 + \tfrac{9}{16}}$ When $B =0.01$ we obtain $A = \boxed{0.7500666637}$

Note that $\frac{d}{dt}A^2 = 2A\dot A = -2AB;\ \ \ \ \ \ \frac{d}{dt}B^2 = 2B\dot B = -2BA$ are equal, so the squares of these functions decrease by the same amount.

Since $B^2$ decreases by $1^2 - 0.01^2 = 0.9999$ , $A^2$ decreases by the same amount, becoming $A^2 = 1.25^2 - 0.9999 = 0.5626,$ so that $A = \sqrt{0.5626} \approx 0.750$ .

By differentiating the first equation, and using the second equation, we get

$A''=-B'=A$

Now, this simple equation can be solved to get

$A\left(x\right)=C_1e^x+C_2e^{-x}$

Here, $\displaystyle C_1\ ,\ C_2$ are arbitrary constants. Putting this function in the first equation, we get

$B\left(x\right)=C_2e^{-x}-C_1e^x$

Now, using the initial values, the constants can be determined easily and they come out to be $\displaystyle C_1=0.125$ and $\displaystyle C_2=1.125$

$\{\{\text{aa},\,\text{bb}\}\}=\{a,b\}\text{/.}\, \text{DSolve}\left[\left\{a'(t)=b(t),b'(t)=a(t),a(0)=1.25,b(0)=1.\right\},\{a,b\},t\right]$

$\text{aa}: \{t\}\to \frac{1}{8} e^{-t} \left(9 e^{2 t}+1\right)$

$\text{bb}: \{t\}\to \frac{1}{8} e^{-t} \left(9 e^{2 t}-1\right)$

There are two families of solutions for $\text{bb}(t)=\frac{1}{100}$ :

$\left\{t\to \text{ConditionalExpression}\left[2 i \pi c_1+i \pi +\log \left(\frac{1}{225} \left(\sqrt{5626}-1\right)\right),c_1\in \mathbb{Z}\right]\right\}$ giving $\frac{225 e^{-2 i \pi c_1-i \pi } \left(1+9 e^{2 \left(2 i \pi c_1+i \pi +\log \left(\frac{1}{225} \left(\sqrt{5626}-1\right)\right)\right)}\right)}{8 \left(\sqrt{5626}-1\right)}$

$\left\{t\to \text{ConditionalExpression}\left[\log \left(\frac{1}{225} \left(\sqrt{5626}+1\right)\right)+2 i \pi c_1,c_1\in \mathbb{Z}\right]\right\}$ giving $\frac{225 e^{-2 i \pi c_1} \left(1+9 e^{2 \left(\log \left(\frac{1}{225} \left(\sqrt{5626}+1\right)\right)+2 i \pi c_1\right)}\right)}{8 \left(\sqrt{5626}+1\right)}$

For $c_1\to 0$ , the expressons, within rounding error, evaluate to $-0.750066663703967$ and $0.750066663703967$ , respectively.

I used the real time solution to respond to the problem.