Let f:(0,∞)→R be a function which is differentiable at all points of its domain and satisfies the condition x2f′(x)=2xf(x)+3, with f(1)=4. Then 2f(2) is equal to:
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Step-by-Step Solution
Step 1: Rearrange the differential equation
The given differential equation is x2f′(x)=2xf(x)+3. To solve this, we first rearrange it into a standard form. We move the term 2x f(x) to the left side of the equation.
Step 2: Divide by x4 to form a derivative of a quotient
The left side of the equation now resembles the numerator of the quotient rule for differentiation, specifically dxd(x2f(x))=x4x2f′(x)−f(x)(2x). By dividing both sides of our rearranged equation by x4, we transform the left side into the derivative of x2f(x).
Step 3: Integrate both sides
Now that the left side is the derivative of a function, we can integrate both sides with respect to x. The integral of a derivative simply gives back the original function, plus a constant of integration.
Step 4: Solve the integrals
We perform the integration on both sides. The integral of dxd(x2f(x)) is x2f(x). The integral of 3x−4 is 3−3x−3, which simplifies to −x31. We add the constant of integration C.
Step 5: Use the initial condition to find C
We are given the initial condition f(1)=4. We substitute x=1 and f(x)=4 into our integrated equation to solve for the constant C. This gives us 4=−1+C, so C=5.
Step 6: Find f(x) and then 2f(2)
Now that we have the value of C, we can write the explicit form of f(x). We then substitute x=2 into this function to find f(2), and finally multiply by 2 to get the required value.