Let f be a differentiable function such that 2(x+2)2f(x)−3(x+2)2=10∫0x(t+2)f(t)dt, x≥0. Then f(2) is equal to
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Step-by-Step Solution
Step 1: Rewrite the given equation
The problem provides an equation involving a differentiable function f(x) and a definite integral. Our goal is to find the value of f(2). First, let's rewrite the given equation to make it easier to differentiate.
Step 2: Differentiate both sides with respect to x
To eliminate the integral, we differentiate both sides of the equation with respect to x. We will use the product rule for differentiation on the left side and the Fundamental Theorem of Calculus on the right side.
Step 3: Apply differentiation rules
Applying the product rule to 2(x+2)2f(x) gives 2[2(x+2)f(x)+(x+2)2f′(x)]. Differentiating −3(x+2)2 gives −6(x+2). By the Fundamental Theorem of Calculus, the derivative of 10∫0x(t+2)f(t)dt is 10(x+2)f(x).
Step 4: Simplify the differential equation
We expand and simplify the equation. Since x≥0, x+2 is never zero, so we can divide the entire equation by 2(x+2) to simplify it further.
Step 5: Rearrange into a linear first-order differential equation
Rearranging the terms, we get a first-order linear differential equation: (x+2)f′(x)−3f(x)=3. This equation can be solved using an integrating factor or by recognizing it as a derivative of a product.
Step 6: Solve the differential equation
We can rewrite the differential equation as (x+2)4(x+2)f′(x)−3f(x)=(x+2)43. The left side is the derivative of (x+2)3f(x). Integrating both sides gives (x+2)3f(x)=−(x+2)31+C.
Step 7: Find the constant C and f(2)
From the integrated form, f(x)=−1+C(x+2)3. To find C, we use the original equation at x=0. Substituting x=0 into the original equation gives 8f(0)−12=0, so f(0)=3/2. Plugging this into our solution for f(x) yields C=5/16. Therefore, f(x)=−1+165(x+2)3. Finally, we calculate f(2) by substituting x=2 into this expression.