3. a. Solve the differential equation y prime equals the product of 4 times x and the square root of the quantity 1 minus y squared
b. Explain why the initial value problem y prime equals the product of 4 times x and the square root of the quantity 1 minus y squared with y(0) = 4 does not have a solution.

(a) It looks like the ODE is

y' = 4x √(1 - y ^2)

which is separable:

dy/dx = 4x √(1 - y ^2)   =>   dy/√(1 - y ^2) = 4x dx

Integrate both sides. On the left, substitute y = sin(t ) and dy = cos(t ) dt :

∫ dy/√(1 - y ^2) = ∫ 4x dx

∫ cos(t ) / √(1 - sin^2(t )) dt = ∫ 4x dx

∫ cos(t ) / √(cos^2(t )) dt = ∫ 4x dx

∫ cos(t ) / |cos(t )| dt = ∫ 4x dx

Since we want the substitutiong to be reversible, we implicitly assume that -π/2 ≤ t ≤ π/2, for which cos(t ) > 0, and in turn |cos(t )| = cos(t ). So the left side reduces completely and we get

∫ dt = ∫ 4x dx

t = 2x ^2 + C

arcsin(y) = 2x ^2 + C

y = sin(2x ^2 + C )

(b) There is no solution for the initial value y (0) = 4 because sin is bounded between -1 and 1.

10/60

step-by-step explanation:

if you divide 10 by 60 the answer is 0.166666 while if you divide or solve the other problems it equals 0.6

step-by-step explanation:

if you want to solve percent error follow these steps.

1. subtract the accepted value from the experimental value.

2. take the absolute value of step 1.

3. divide that answer by the accepted value.

4. multiply that answer by 100 and add the % symbol to express the answer as a percentage.

if you want to solve percent of change just do this.

1. work out the difference (increase) between the two numbers you are comparing.

2. divide the increase by the original number and multiply the answer by 100. if your answer is a negative number then this is a percentage decrease.

hope this : )