Differential Equations

Question

1. a) Solve the differential equation y' = 4x√(1-y²).
b) Explain why the initial value problem y' = 4x√(1-y²) with y(0) = 4 does not have a solution.

 

2. Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for Int [0,4] x²dx.

Answer 1

1.

a)y' = 4x√(1-y²)

dy/√(1-y²)=4xdx

arcsin(y)=2x²+C

y=sin(2x²+C)

 

b) y(0)=4

4=sin(C). It cannot happen because |sin(x)|≤1