Question

Find the particular solution of the differential equation \( \frac{dy}{dx} = \frac{y}{x} \) given the initial boundary condition that \( y = 2 \) when \( x = 1 \).

• A. \( y = 2x \)
• B. \( y = x + 1 \)
• C. \( y = x^2 \)
• D. \( y = \frac{2}{x} \)

Hint:

When every term in a separated integral results in a natural logarithm, writing your integration constant as \( \ln C \) instead of just \( C \) makes simplifying the logarithms much cleaner.

Answer 1

The Correct Option is A

Concept: A particular solution is found by first separating variables and integrating to determine the general solution with an integration constant \( C \), and then substituting the initial boundary conditions to solve for the exact value of that constant.

Step 1: Separate the variables and integrate both sides. Group the \( y \) variables on the left and the \( x \) variables on the right: \[ \frac{1}{y}\,dy = \frac{1}{x}\,dx \] Set up and evaluate the integrals: \[ \int \frac{1}{y}\,dy = \int \frac{1}{x}\,dx \quad \Rightarrow \quad \ln|y| = \ln|x| + \ln|C| \] Using logarithmic properties to combine the right side: \[ \ln|y| = \ln|Cx| \quad \Rightarrow \quad y = Cx \]

Step 2: Substitute the boundary conditions to find the constant \( C \). We are given that \( y = 2 \) when \( x = 1 \). Substitute these coordinates into our general solution: \[ 2 = C \cdot (1) \quad \Rightarrow \quad C = 2 \]

Step 3: Write out the final particular solution equation. Replace the constant \( C \) with its calculated value of 2 in our general equation: \[ y = 2x \]