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, $$f(t)=\frac{e^{2t}}{1-t}\mbox{ and }g(t)=e^{2t}$$