What is the particular solution to the differential equation (dy)/(dx)=(y)/(x) with the initial condition y(e^(2))=-1 ?

Identify type of differential equation: Step 1: Identify the type of differential equation. We have (dy)/(dx) = (y)/(x), which is a separable differential equation.Separate variables: Step 2: Separate the variables. Rearrange to get dy/y = dx/x.Integrate both sides: Step 3: Integrate both sides. Integrate dy/y to get ln|y| and integrate dx/x to get ln|x| + C, where C is the constant of integration.Solve for y: Step 4: Solve for y. Exponentiate both sides to solve for y: y = e^(ln|x| + C) = |x| * e^C. Let A = e^C, then y = A|x|.Apply initial condition: Step 5: Apply the initial condition to find A. Substitute x = e^2 and y = -1 into y = A|x| to get -1 = A*e^2.Solve for A: Step 6: Solve for A. A = -1/e^2.Write particular solution: Step 7: Write the particular solution. The particular solution is y = (-1/e^2)|x|.

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Q. What is the particular solution to the differential equation

\frac{d y}{d x}=\frac{y}{x}

with the initial condition

y\left(e^{2}\right)=-1

Identify type of differential equation: Step $1$ : Identify the type of differential equation. $\newline$ We have $\frac{dy}{dx} = \frac{y}{x}$ , which is a separable differential equation.
Separate variables: Step $2$ : Separate the variables. $\newline$ Rearrange to get $\frac{dy}{y} = \frac{dx}{x}$ .
Integrate both sides: Step $3$ : Integrate both sides. $\newline$ Integrate $\frac{dy}{y}$ to get $\ln|y|$ and integrate $\frac{dx}{x}$ to get $\ln|x| + C$ , where $C$ is the constant of integration.
Solve for y: Step $4$ : Solve for y. $\newline$ Exponentiate both sides to solve for y: $y = e^{\ln|x| + C} = |x| \cdot e^C$ . Let $A = e^C$ , then $y = A|x|$ .
Apply initial condition: Step $5$ : Apply the initial condition to find $A$ . Substitute $x = e^2$ and $y = -1$ into $y = A|x|$ to get $-1 = A\cdot e^2$ .
Solve for A: Step $6$ : Solve for A. $\newline$ $A = -\frac{1}{e^2}$ .
Write particular solution: Step $7$ : Write the particular solution. $\newline$ The particular solution is $y = \frac{-1}{e^{2}}|x|$ .