Differential Equations Question - None

Question

If $y = f(x)$ satisfy the differential equation $\frac{dy}{dx} + \frac{y}{x} = x^2$; $f(1) = 1$; then value of $f(3)$ equals:

Accepted Answer

Step 1: Identify the form

The equation is: dy/dx + y/x = x²

This is a linear ODE of the form dy/dx + P(x)y = Q(x) where P(x) = 1/x and Q(x) = x²

Step 2: Find the integrating factor

μ(x) = e^(∫P(x)dx) = e^(∫(1/x)dx) = e^(ln|x|) = x

Step 3: Multiply the equation by μ(x)

x·(dy/dx) + x·(y/x) = x·x²

x(dy/dx) + y = x³

Step 4: Recognize the left side as a derivative

d/dx(xy) = x³

Step 5: Integrate both sides

∫d/dx(xy)dx = ∫x³dx

xy = x⁴/4 + C

Step 6: Apply the initial condition f(1) = 1

When x = 1, y = 1:

1·1 = (1)⁴/4 + C

1 = 1/4 + C

C = 3/4

Step 7: Write the particular solution

xy = x⁴/4 + 3/4

y = x³/4 + 3/(4x)

Step 8: Find f(3)

f(3) = (3)³/4 + 3/(4·3) = 27/4 + 3/12 = 27/4 + 1/4 = 28/4 = 7

∴ Answer: a

Question: