Final answer:
To solve the differential equation dy/dx = sqrt(y² - 25), we can start by separating variables and integrating both sides. The final solution is y = sqrt(25 + sin^2(2x + C)).
Step-by-step explanation:
To solve the differential equation dy/dx = sqrt(y² - 25), we can start by separating variables. Rearranging the equation, we have dy/sqrt(y² - 25) = dx. Next, we integrate both sides with respect to their corresponding variables. On the left side, we can use the substitution u = y² - 25, which gives 1/2∫du/sqrt(u) = x + C, where C is the constant of integration. Simplifying, we obtain arcsin(sqrt(u)) = 2x + C. Finally, replacing u with y² - 25 and solving for y, we have y = sqrt(25 + sin^2(2x + C)).