I have the following system:
$f(t) = t \cdot f(t) + g(t)$ $g'(t) = g(t) + t \cdot g'(t) + f(t)$
which I want to solve for $f(t)$ and $g(t)$. I also have initial conditions that $f(0)=g(0)=1$. How do I go about this in the best way?
I have the following system:
$f(t) = t \cdot f(t) + g(t)$ $g'(t) = g(t) + t \cdot g'(t) + f(t)$
which I want to solve for $f(t)$ and $g(t)$. I also have initial conditions that $f(0)=g(0)=1$. How do I go about this in the best way?
From the first equation, $g(t)=(1-t)f(t)\Rightarrow g^{\prime}(t)=(1-t)f^\prime (t)-f(t)$Now from the second, $(1-t)g^\prime (t)=g(t)+f(t)\Rightarrow (1-t)^2f^\prime (t)-(1-t)f(t)=(1-t)f(t)+f(t)$ So we have$(1-t)^2f^\prime (t)=(3-2t)f(t)$ Hence $\frac{f^\prime (t)}{f(t)}=\frac{3-2t}{(1-t)^2}$ Now integrate both sides to get, $\ln (f(t))=\int\frac{3-2t}{(1-t)^2}dt+C=\int \frac{1+2(1-t)}{1-t}dt+C=-\ln (1-t)+2t+C$ and now using initial condition, $C=0.$ Hence, $f(t)=\frac{e^{2t}}{1-t}\mbox{ and }g(t)=e^{2t}$