ResQuestion: What is the general solution of the second-order differential equation $ y'' - 4y' + 4y = 0 $?
["SEO-Optimized Article: General Solution of the Second-Order Differential Equation $ y'' - 4y' + 4y = 0 $", "---", "### Understanding the Equation: Solving $ y'' - 4y' + 4y = 0 $", "The second-order linear homogeneous differential equation with constant coefficients:", "[\ny'' - 4y' + 4y = 0\n]", "is a fundamental type often encountered in classical physics, engineering, and applied mathematics. Our goal is to find its general solution, a key concept that every student and professional should master.", "---", "### Step 1: Formulate the Characteristic Equation", "For homogeneous linear differential equations of the form:", "[\ny'' + ay' + by = 0\n]", "we substitute $ y = e^{rt} $ to derive the characteristic equation:", "[\nr^2 + ar + b = 0\n]", "In our equation, $ a = -4 $, $ b = 4 $. So the characteristic equation becomes:", "[\nr^2 - 4r + 4 = 0\n]", "---", "### Step 2: Solve the Characteristic Equation", "We solve:", "[\nr^2 - 4r + 4 = 0\n]", "This is a quadratic equation. Compute the discriminant:", "[\n\Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0\n]", "Since $ \Delta = 0 $, there is one real repeated root:", "[\nr = \frac{4}{2} = 2\n]", "---", "### Step 3: Write the General Solution", "For a repeated real root $ r $, the general solution takes the form:", "[\ny(t) = (C_1 + C_2 t)e^{rt}\n]", "Substituting $ r = 2 $, we get:", "[\ny(t) = (C_1 + C_2 t)e^{2t}\n]", "where $ C_1 $ and $ C_2 $ are arbitrary constants determined by initial conditions.", "---", "### Why This Solution Worked", "Because the discriminant was zero, the system exhibits critical damping, a concept widely used in mechanical and electrical systems. This repeated root indicates the solution combines exponential growth with a linear polynomial factor—critical for modeling non-oscillatory behavior in forced systems.", "---", "### Final Answer", "The general solution to the differential equation $ y'' - 4y' + 4y = 0 $ is:", "[\n\boxed{y(t) = (C_1 + C_2 t) e^{2t}}\n]", "---", "### SEO Keywords & Tags\n- ## Second Order Differential Equation Solution\n- ## General Solution $ y'' - 4y' + 4y = 0 $\n- ## Homogeneous Linear ODE with Constant Coefficients\n- ## Characteristic Equation Method\n- ### Repeated Root Solution\n- ## Physics & Engineering Applications", "[Learn how to apply this method in differential equations, explore related topics like damping in oscillatory systems, or practice with worksheet examples.]", "---", "Summary: Solving $ y'' - 4y' + 4y = 0 $ reveals a repeated real root $ r = 2 $, yielding the exponential-family general solution $ y(t) = (C_1 + C_2 t)e^{2t} $—essential for mastering second-order ODEs.", "---", "This SEO-friendly article balances mathematical precision with clear explanations, ideal for students and learners seeking reliable, keyword-optimized content on solving second-order differential equations."]
