Question

For t≥0 a parametric curve is described by the following equations

dx/dt = 16t²/3x and y = 1/2t²-2t and passes through x=0 when t=0. a) Using separation of variables, solve the ODE for x(t) to obtain an expression for x in terms of t.

Answer 1

To solve the equation dx/dt = (16t^2/3)x using separation of variables, integrate both sides of the equation and solve for x.

Step-by-step explanation:

To solve the equation dx/dt = (16t^2/3)x, we can use separation of variables. Start by rewriting the equation as dx/x = (16t^2/3)dt. Now, integrate both sides of the equation: ∫(1/x)dx = ∫(16t^2/3)dt. The integral of 1/x is ln|x|, and the integral of (16t^2/3) is (16/9)t^3 + C, where C is the constant of integration. So we have ln|x| = (16/9)t^3 + C. Solving for x, we can take the exponential of both sides to eliminate the ln function: x = e^((16/9)t^3 + C).