Question Statement

Solve the following first-order differential equation:

$x\frac{dy}{dx}=4y$

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Background and Explanation

This is a first-order ordinary differential equation (ODE) that can be solved using the separation of variables method. This technique involves rearranging the equation so that all terms involving $y$ are on one side and all terms involving $x$ are on the other before integrating.

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Solution

To solve the equation, we follow these steps:

Step 1: Separate the variables

We begin by moving all $y$ terms to the left side and all $x$ terms to the right side.

Starting with: $x\frac{dy}{dx}=4y$

Multiply both sides by $dx$: $xdy=4ydx$

Divide both sides by $y$ and $x$ to isolate the variables: $\frac{dy}{y}=\frac{4}{x}dx$

Step 2: Integrate both sides

Now that the variables are separated, we apply the integral to both sides of the equation:

$\int \frac{dy}{y}=4\int \frac{1}{x}dx$

Performing the integration gives us: $\ln y=4\ln x+\ln c$

Note: We use $\ln c$ as the constant of integration to make the subsequent algebraic simplification easier.

Step 3: Simplify using logarithmic properties

We can use the power rule of logarithms ($n\ln a=\ln a^n$) to rewrite the right side: $\ln y=\ln x^4+\ln c$

Next, use the product rule of logarithms ($\ln a+\ln b=\ln (ab)$): $\ln y=\ln \left(c{x}^4\right)$

Step 4: Solve for $y$

By taking the exponential of both sides (or simply removing the natural logs), we find the general solution: $y=c{x}^4$

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Key Formulas or Methods Used

• Separation of Variables: Rearranging $\frac{dy}{dx}=f(x)g(y)$ into $\frac{1}{g(y)}dy=f(x)dx$.
• Logarithmic Integration: $\int \frac{1}{u}du=\ln |u|+C$.
• Logarithm Power Rule: $n\ln (x)=\ln (x^n)$.
• Logarithm Product Rule: $\ln (a)+\ln (b)=\ln (ab)$.

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Summary of Steps

1. Rearrange the differential equation to group $y$ terms with $dy$ and $x$ terms with $dx$.
2. Integrate both sides of the equation.
3. Add a constant of integration (expressed as $\ln c$ for convenience).
4. Apply logarithmic identities to combine terms on the right side.
5. Exponentiate both sides to solve for the explicit function $y(x)$.