Question

Provide the explicit solution to the differential equation
dt/dx=(x²)-4x

Answer 1

To find the explicit solution to the differential equation dt/dx = x² - 4x, we can separate variables and integrate both sides. The explicit solution is t = (1/3)x³ - 2x² + C, where C is the constant of integration.

Step-by-step explanation:

To find the explicit solution to the differential equation dt/dx = x² - 4x, we can separate variables and integrate both sides. Rearranging the equation, we have:

dt = (x² - 4x)dx

Integrating both sides:

∫dt = ∫(x² - 4x)dx

Integrating the left side gives t = t + C1, where C1 is the constant of integration. For the right side:

∫(x² - 4x)dx = (∫x²dx) - (∫4xdx) = (1/3)x³ - 2x² + C2, where C2 is another constant of integration.

Combining the two sides, we have the explicit solution to the differential equation: t = (1/3)x³ - 2x² + C, where C = C2 - C1 is the final constant of integration.