Question

Solve the differential equation. There are two solutions. One solution is given. (Note: The constant of integration, C, will appear in your answer. Hint/Tip: since 1 - t^2 is always positive, you do not need to use absolute values after integrating.)

(1 - t^2)y'
a) (1 - t^2)y = C
b) (1 + t^2)y = C
c) (1/t^2)y = C
d) (1 - 2t)y = C

Answer 1

To solve the differential equation (1 - t^2)y', we use separation of variables and integration, ultimately obtaining the general solution, (1 - t^2)y = C, which matches option (a).

Step-by-step explanation:

The differential equation in question is (1 - t^2)y'. The task is to solve for y. Two solutions are presented: (a) (1 - t^2)y = C, and (b) a different equation from the choices provided. To find the correct differential equation for y, we integrate both sides of the original differential equation concerning t. The separation of variables leads to dy/y = dt/(1 - t^2). We integrate both sides to get ln(|y|) = 1/2 ln(1 - t^2) + C, and exponentiate both sides to solve for y, yielding y = ±e^(C)(1 - t^2)^(1/2), where C is the constant of integration. Since we are instructed that 1 - t^2 is always positive, we can ignore the absolute values. Hence, our general solution is (1 - t^2)y = C. Our final solution matches option (a).