6. the function ( y = f(x) ) satisfies the differential equation ( \frac{dy}{dx} = 2xy ) with the initial…

Explanation:

Step1: Separate variables

Given (\frac{dy}{dx}=2xy), we can rewrite it as (\frac{dy}{y}=2x dx) (separate (y) and (x) terms).

Step2: Integrate both sides

Integrate (\int\frac{dy}{y}=\int 2x dx). Using the integral formulas (\int\frac{1}{u}du=\ln|u| + C) ((u = y)) and (\int ax^{n}dx=\frac{ax^{n + 1}}{n+1}+C) ((a = 2), (n=1)), we get (\ln|y|=x^{2}+C).

Step3: Solve for (y)

Exponentiating both sides, (y = e^{x^{2}+C}=e^{C}e^{x^{2}}). Let (k = e^{C}), so (y=ke^{x^{2}}).

Step4: Use the initial condition

Given (f(0)=5), substitute (x = 0) and (y = 5) into (y=ke^{x^{2}}). We have (5=ke^{0}), since (e^{0}=1), then (k = 5). So the function is (y = 5e^{x^{2}}).

Step5: Find (f(2))

Substitute (x = 2) into (y = 5e^{x^{2}}). Then (y=5e^{2^{2}}=5e^{4}).

Answer:

(5e^{4})