Question

The solution of the differential equation y′=x²y is

A)y=ceˣ²
B)y=c+eˣ³/³
C)y=c+eˣ²
D)y=ceˣ³
E)y=ceˣ³/³

Answer 1

The solution of the differential equation y' = x²y is y = ce^((1/3)x³).

Step-by-step explanation:

The differential equation y' = x²y is a first-order linear homogeneous differential equation. To solve it, we'll use the method of separation of variables. Here are the steps:

1. Separate the variables by multiplying both sides of the equation by dx:
2. y' dx = x²y dx
3. Divide both sides by y:
4. (1/y) dy = x² dx
5. Integrate both sides:
6. ln|y| = (1/3)x³ + C
7. Exponentiate both sides:
8. |y| = e^((1/3)x³ + C)
9. Remove the absolute value:
10. y = ±e^((1/3)x³ + C)
11. Combine the constant of integration:
12. y = ce^((1/3)x³)

So the solution to the differential equation is y = ce^((1/3)x³). Therefore, the correct answer is D) y = ce^((1/3)x³).