8 Multiple Choice 1 point Solve the differential equation: $$ f^{\prime \prime}(x)=x^{-3 / 2}, f^{\prime}(4)=2, f(0)=0 $$

Question

UpStudy AI · Accepted Answer

We start with the differential equation

f″(x) = x^(–3/2).

Step 1. Integrate f″(x) to find f′(x):

f′(x) = ∫x^(–3/2) dx.

Recall that ∫x^(n) dx = x^(n+1)/(n+1) + constant, provided n ≠ –1. Here, n = –3/2, so

n + 1 = (–3/2) + 1 = –1/2.

Thus,
  ∫x^(–3/2) dx = x^(–1/2)/(–1/2) + C₁ = –2x^(–1/2) + C₁.

So,
  f′(x) = –2/√x + C₁.

Step 2. Use the condition f′(4) = 2:

f′(4) = –2/√4 + C₁ = –2/2 + C₁ = –1 + C₁ = 2.

So, C₁ = 3, and therefore,
  f′(x) = –2/√x + 3.

Step 3. Integrate f′(x) to find f(x):

f(x) = ∫[–2/√x + 3] dx = –2∫x^(–1/2) dx + 3∫dx.

We compute the integrals:
  ∫x^(–1/2) dx = 2√x,
  ∫dx = x.

Thus,
  f(x) = –2(2√x) + 3x + C₂ = –4√x + 3x + C₂.

Step 4. Use the condition f(0) = 0:

Since √0 = 0, we have
  f(0) = –4(0) + 3(0) + C₂ = C₂ = 0.

Thus, the constant C₂ = 0.

Final answer:
  f(x) = 3x – 4√x.

Multiple Choice Answer: f(x) = 3x – 4√x.
The solution to the differential equation is $$ f(x) = 3x - 4\sqrt{x} $$.

8 Multiple Choice 1 point Solve the differential equation: \( f^{\prime \prime}(x)=x^{-3 / 2}, f^{\prime}(4)=2, f(0)=0 \)