find the solution of the differential equation that satisfies the given initial condition…

Explanation:

Step1: Separate variables

Separate the variables in the differential equation (\frac{dr}{dq}=q^{2}r). We get (\frac{dr}{r}=q^{2}dq).

Step2: Integrate both sides

Integrate (\int\frac{dr}{r}=\int q^{2}dq). Using the integral formulas (\int\frac{1}{x}dx=\ln|x| + C) and (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)), we have (\ln|r|=\frac{q^{3}}{3}+C).

Step3: Solve for (r)

Exponentiate both sides to get (r = e^{\frac{q^{3}}{3}+C}=e^{C}e^{\frac{q^{3}}{3}}). Let (A = e^{C}), so (r=Ae^{\frac{q^{3}}{3}}).

Step4: Use the initial condition

Use the initial condition (r(3)=5). Substitute (q = 3) and (r = 5) into (r=Ae^{\frac{q^{3}}{3}}). We have (5=Ae^{\frac{3^{3}}{3}}), which simplifies to (5=Ae^{9}). Then (A = 5e^{-9}).

Answer:

(r = 5e^{\frac{q^{3}}{3}-9})