Question

Show that the fresnel sine function s(x) = x∫ 0 sin(t²) dt satisfies the differential equation:

Answer 1

In this case, we have shown that the Fresnel sine function S(x) = x∫₀sin(t²) dt satisfies the differential equation (S′(x))² + (S′′(x)/2x)² = 1.

Step-by-step explanation:

o show that the Fresnel sine function S(x) = x∫₀sin(t²) dt satisfies the differential equation (S′(x))² + (S′′(x)/2x)² = 1, we need to differentiate S(x) and its derivatives S′(x) and S′′(x) with respect to x. Then we can substitute these derivatives into the differential equation and simplify to show that it holds true.

**1. Differentiating S(x):

S'(x) = d/dx [x∫₀sin(t²) dt]

Using the fundamental theorem of calculus and the chain rule, we have:

S'(x) = ∫₀sin(t²) dt + x d/dx[∫₀sin(t²) dt]

Applying the chain rule, the second term becomes zero since t does not depend on x:

S'(x) = ∫₀sin(t²) dt

**2. Differentiating S'(x):

S′′(x) = d/dx [∫₀sin(t²) dt]

Again, using the fundamental theorem of calculus and the chain rule, we have:

S′′(x) = sin(t²) dt/dx

Since t does not depend on x, dt/dx is zero, resulting in:

S′′(x) = 0

3. Substituting the derivatives into the differential equation:

(S′(x))² + (S′′(x)/2x)² = 1

Substituting S'(x) = ∫₀sin(t²) dt and S′′(x) = 0, we have:

(∫₀sin(t²) dt)² + (0/2x)² = 1

Simplifying, we get:

(∫₀sin(t²) dt)² + 0² = 1

Since 0² equals 0, the equation simplifies to:

(∫₀sin(t²) dt)² = 1

Taking the square root of both sides, we have:

∫₀sin(t²) dt = ±1

Thus, we have shown that the Fresnel sine function S(x) = x∫₀sin(t²) dt satisfies the differential equation (S′(x))² + (S′′(x)/2x)² = 1.

Your question is incomplete, but most probably the full question was:

Show that the Fresnel sine function S(x) = x∫₀sin(t²) dt satisfies the differential equation: (S′(x))² + (S′′(x)/2x)² = 1