Question What is the particular solution to the differential equation (dy)/(dx)=(1+y)/(x) with the initial condition y(e)=1?

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Question What is the particular solution to the differential equation  (dy)/(dx)=(1+y)/(x) with the initial condition  y(e)=1?

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Identify type of differential equation:  Step 1: Identify the type of differential equation. We have (dy)/(dx) = (1+y)/(x). This is a first-order linear differential equation. Rearrange to separate variables:  Step 2: Rearrange the equation to separate variables. (dy)/(1+y) = (dx)/x. This step involves separating the variables y and x on different sides of the equation. Integrate both sides:  Step 3: Integrate both sides. Integrate left side: ∫(dy)/(1+y) = ln|1+y| + C1. Integrate right side: ∫(dx)/x = ln|x| + C2. Combine constants and solve:  Step 4: Combine the constants and solve for y. ln|1+y| = ln|x| + C, where C = C2 - C1. Remove the absolute values (assuming x and 1+y are positive): 1+y = e^(ln|x| + C) = x*e^C. Apply initial condition:  Step 5: Apply the initial condition to find C. Given y(e) = 1, plug in x = e: 1 + 1 = e*e^C. 2 = e^(1+C). Taking natural log on both sides: ln(2) = 1 + C. C = ln(2) - 1. Write particular solution:  Step 6: Write the particular solution using the value of C. y = x*e^(ln(2)-1) - 1. Simplify: y = x*(2/e) - 1.

Question $\newline$ What is the particular solution to the differential equation $\frac{d y}{d x}=\frac{1+y}{x}$ with the initial condition $y(e)=1 ?$

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Question $\newline$ What is the particular solution to the differential equation $\frac{d y}{d x}=\frac{1+y}{x}$ with the initial condition $y(e)=1 ?$