Solve the differential equation \( rac{dy}{dx} = 3y \) with initial condition \( y(0) = 2 \).
["## Solving the Differential Equation ( \frac{dy}{dx} = 3y ) with Initial Condition ( y(0) = 2 )", "Differential equations are fundamental tools in science, engineering, and mathematics, describing how quantities change over time or space. One of the most common and foundational first-order differential equations is the separable equation of the form:", "[\n\frac{dy}{dx} = ky\n]", "where ( k ) is a constant. Today, we will solve the specific differential equation:", "[\n\frac{dy}{dx} = 3y\n]", "with the initial condition:", "[\ny(0) = 2\n]", "This equation models exponential growth or decay—here, since the coefficient is positive, it represents exponential growth. We’ll walk through solving it step-by-step, explaining key concepts and applications.", "---", "### Step 1: Separating Variables", "The given equation:", "[\n\frac{dy}{dx} = 3y\n]", "is separable. To solve it, divide both sides by ( y ) (assuming ( y <br/>\neq 0 )) and multiply by ( dx ):", "[\n\frac{1}{y} , dy = 3 , dx\n]", "This rearrangement isolates variables on opposite sides of the equation.", "---", "### Step 2: Integrating Both Sides", "Now integrate both sides:", "[\n\int \frac{1}{y} , dy = \int 3 , dx\n]", "The integrals yield:", "[\n\ln |y| = 3x + C\n]", "where ( C ) is the constant of integration.", "---", "### Step 3: Solving for ( y )", "To eliminate the logarithm, exponentiate both sides:", "[\n|y| = e^{3x + C} = e^C \cdot e^{3x}\n]", "Let ( A = e^C ), which is a positive constant. Since ( y ) represents a physical quantity and we are given ( y(0) = 2 > 0 ), we can drop the absolute value:", "[\ny = A e^{3x}\n]", "---", "### Step 4: Applying the Initial Condition", "Use the initial condition ( y(0) = 2 ) to solve for ( A ):", "[\n2 = A e^{3 \cdot 0} = A \cdot 1 \Rightarrow A = 2\n]", "---", "### Step 5: Final Solution", "Substitute ( A = 2 ) back into the solution:", "[\ny(x) = 2e^{3x}\n]", "This is the unique solution to the differential equation satisfying the given initial condition.", "---", "### Interpretation and Applications", "The function ( y(x) = 2e^{3x} ) describes exponential growth with a continuous rate of 3. Applications include modeling bacterial growth, radioactive decay (when the coefficient is negative), financial interest accumulation, and population dynamics.", "---", "### Summary", "The solution to the differential equation:", "[\n\frac{dy}{dx} = 3y, \quad y(0) = 2\n]", "is:", "[\n\boxed{y(x) = 2e^{3x}}\n]", "This result demonstrates the power of separable differential equations and their role in modeling real-world phenomena involving continuous change.", "---", "### Key Takeaways", "- Differential equations describe rates of change.\n- Separation of variables is a powerful technique for solving simple first-order ODEs.\n- Initial conditions determine unique solutions.\n- The solution ( y(x) = 2e^{3x} ) shows rapid exponential growth.\n- Understanding such equations is essential in science and engineering.", "For further exploration, try solving related equations like ( \frac{dy}{dx} = ky ) with different ( k ), or consider them in applied contexts such as compound interest or physics.", "---", "Keywords: solve ( \frac{dy}{dx} = 3y ), differential equation, exponential growth, separation of variables, initial condition, ( y(0) = 2 ), ( y(x) = 2e^{3x} )"]
