Tag: integrating factor proof

The Ultimate Freshman Calculus Guide to Differential Equations - Good Grade This Way Freshman Calculus Differential Equations: The Ultimate Guide Your Ultimate Guide to First-Order separable, Linear Equations, and Modeling In standard freshman calculus, you spend months learning how to calculate the derivative \(f'(x)\) of a known function \(f(x)\), or how to integrate a rate to find the net change. However, in the real world, nature rarely gives us functions directly. Instead, physical laws dictate how things change. We are given relationships between a changing quantity and its current state. To unlock the functions themselves, we use differential equations. 1. What is a Differential Equation? Conceptually, a differential equation (DE) is any equation that relates an unknown function to one or more of its derivatives. Solving a differential equation means finding a function—or a family of functions—that satisfies this relationship. Key Classification Terms Ordinary Differential Equation (ODE): An equation involving derivatives with respect to only a single independent variable (the focus of freshman calculus). Order: The order of the highest derivative present in the equation. Degree: The power to which the highest derivative is raised (after removing fractional powers). \(x^2 \displaystyle\frac{d^2y}{dx^2} + 3\displaystyle\frac{dy}{dx} + y = e^x \quad \text{is a second-order, first-degree ODE.}\) General vs. Particular Solutions Because integration naturally introduces an arbitrary constant \(C\), solving a differential equation yields an infinite family of functions called the general solution. When we are provided with an initial condition (e.g., \(y(x_0) = y_0\)), we can solve for \(C\) to secure a unique particular solution. This setup is called an Initial Value Problem (IVP). 2. First-Order Separable Equations The first major class of differential equations you will learn to solve is separable equations. These are equations that can be algebraically factored so that all terms involving \(y\) can be grouped on one side, and all terms involving \(x\) can be grouped on the other. Separable Form and Integration Blueprint Any first-order separable equation can be written in the form: \(\displaystyle\frac{dy}{dx} = g(x)h(y)\) If \(h(y) \neq 0\), we separate the variables by dividing by \(h(y)\) and multiplying by the differential \(dx\): \(\displaystyle\frac{1}{h(y)} \, dy = g(x) \, dx \to \int \displaystyle\frac{1}{h(y)} \, dy = \int g(x) \, dx\) Example 1: Solving a Basic Separable Equation Find the general solution of the differential equation: \(\displaystyle\frac{dy}{dx} = \displaystyle\frac{x^2}{y}\) Step 1: Separate the variables Multiply both sides by \(y\) and \(dx\) to group like-variables: \(y \, dy = x^2 \, dx\) Step 2: Integrate both sides \(\int y \, dy = \int x^2 \, dx \to \displaystyle\frac{1}{2}y^2 = \displaystyle\frac{1}{3}x^3 + C_1\) Step 3: Isolate the dependent variable \(y\) Multiply the entire equation by \(2\), letting \(2C_1 = C\) represent a new arbitrary constant: \(y^2 = \displaystyle\frac{2}{3}x^3 + C \to y = \pm\sqrt{\displaystyle\frac{2}{3}x^3 + C}\) \(y = \pm\sqrt{\displaystyle\frac{2}{3}x^3 + C}\) Example 2: Separable Initial Value Problem (IVP) Solve the initial value problem: \(\displaystyle\frac{dy}{dx} = (1 + y^2)e^x, \quad y(0) = 1\) Step 1: Separate variables and integrate \(\displaystyle\frac{1}{1 + y^2} \, dy = e^x \, dx \to \int \displaystyle\frac{1}{1 + y^2} \, dy = \int e^x \, dx\) \(\arctan(y) = e^x + C\) Step 2: Apply the initial condition to solve for \(C\) Substitute \(x = 0\) and \(y = 1\): \(\arctan(1) = e^0 + C \to \displaystyle\frac{\pi}{4} = 1 + C \to C = \displaystyle\frac{\pi}{4} - 1\) Step 3: Substitute \(C\) back and isolate \(y\) \(\arctan(y) = e^x + \displaystyle\frac{\pi}{4} - 1 \to y = \tan\left(e^x + \displaystyle\frac{\pi}{4} - 1\right)\) \(y = \tan\left(e^x + \displaystyle\frac{\pi}{4} - 1\right)\) 3. First-Order Linear Equations & The Integrating Factor If a first-order equation is not separable, it might be linear. First-order linear differential equations are solved using a powerful, structured technique called the Integrating Factor Method. Standard Form of a First-Order Linear Equation To solve a linear DE, you must first write it in standard form: \(\displaystyle\frac{dy}{dx} + P(x)y = Q(x)\) Where \(P(x)\) and \(Q(x)\) are continuous functions of \(x\). Mathematical Proof of the Integrating Factor: We want to find a multiplier function, \(I(x)\), that we can distribute across the standard form equation so that the left-hand side simplifies perfectly into a single Product Rule derivative: \(I(x)\displaystyle\frac{dy}{dx} + I(x)P(x)y = \displaystyle\frac{d}{dx}[I(x)y]\) Expanding the right-hand side using the Product Rule yields: \(I(x)\displaystyle\frac{dy}{dx} + I'(x)y\) Equating the coefficients of \(y\) on both sides reveals the required condition for \(I(x)\): \(I'(x) = I(x)P(x)\) This is a separable differential equation in terms of \(I\)! Let's solve it: \(\displaystyle\frac{dI}{dx} = I P(x) \to \displaystyle\frac{1}{I} \, dI = P(x) \, dx \to \ln|I| = \int P(x) \, dx \to I(x) = e^{\int P(x) \, dx}\) The Integrating Factor Formula \(I(x) = e^{\int P(x) \, dx}\) Multiplying both sides of our standard equation by \(I(x)\) allows us to condense and integrate directly: \(\displaystyle\frac{d}{dx}[I(x)y] = I(x)Q(x) \to I(x)y = \int I(x)Q(x) \, dx \to y = \displaystyle\frac{1}{I(x)}\int I(x)Q(x) \, dx\) Example 3: First-Order Linear Differential Equation Find the general solution for \(x > 0\) of the differential equation: \(x \displaystyle\frac{dy}{dx} + 2y = x^3\) Step 1: Convert to Standard Form Divide the entire equation by \(x\) to isolate \(\displaystyle\frac{dy}{dx}\): \(\displaystyle\frac{dy}{dx} + \displaystyle\frac{2}{x}y = x^2\) Here, we identify \(P(x) = \displaystyle\frac{2}{x}\) and \(Q(x) = x^2\). Step 2: Calculate the Integrating Factor \(I(x)\) \(I(x) = e^{\int \frac{2}{x} \, dx} = e^{2\ln|x|} = e^{\ln(x^2)} = x^2 \quad (\text{since } x > 0)\) Step 3: Multiply the standard form by \(I(x)\) and integrate \(x^2 \left(\displaystyle\frac{dy}{dx} + \displaystyle\frac{2}{x}y\right) = x^2(x^2) \to \displaystyle\frac{d}{dx}[x^2 y] = x^4\) Integrate both sides with respect to \(x\): \(x^2 y = \int x^4 \, dx \to x^2 y = \displaystyle\frac{1}{5}x^5 + C\) Step 4: Isolate \(y\) \(y = \displaystyle\frac{1}{5}x^3 + \displaystyle\frac{C}{x^2}\) \(y = \displaystyle\frac{1}{5}x^3 + \displaystyle\frac{C}{x^2}\) 4. Physical Applications and Modeling Differential equations are the language of physics, biology, and chemistry. Two classic modeling templates you must master are Exponential Growth/Decay and Newton's Law of Cooling. Exponential Growth and Decay When a quantity \(y\) grows or decays at a rate directly proportional to its current size, it is modeled by: \(\displaystyle\frac{dy}{dt} = ky\) Separating variables and integrating yields the famous exponential model: \(y(t) = y_0 e^{kt}\), where \(y_0\) is the initial value at \(t = 0\). Newton's Law of Cooling Newton's Law of Cooling states that the rate of change of an object's temperature \(T\) is proportional to the difference between its temperature and the surrounding medium's temperature \(T_s\). Newton's Law of Cooling Equation \(\displaystyle\frac{dT}{dt} = k(T - T_s)\) Where \(k\) is a constant of proportionality. Example 4: Modeling Newton's Law of Cooling A hot cup of coffee at \(95^\circ\text{C}\) is placed in a room kept at a constant temperature of \(20^\circ\text{C}\). After \(10\) minutes, the coffee cools down to \(65^\circ\text{C}\). Find the temperature of the coffee after \(20\) minutes. Step 1: Set up the model With \(T_s = 20\), our differential equation is: \(\displaystyle\frac{dT}{dt} = k(T - 20)\) Step 2: Separate variables and solve \(\displaystyle\frac{1}{T - 20} \, dT = k \, dt \to \ln|T - 20| = kt + C_1 \to T(t) - 20 = C e^{kt}\) \(T(t) = 20 + C e^{kt}\) Step 3: Apply the initial condition to find \(C\) At \(t = 0\), \(T(0) = 95\): \(95 = 20 + C e^0 \to C = 75 \to T(t) = 20 + 75 e^{kt}\) Step 4: Use the second condition to find \(k\) At \(t = 10\), \(T(10) = 65\): \(65 = 20 + 75 e^{10k} \to 45 = 75 e^{10k} \to e^{10k} = \displaystyle\frac{3}{5} = 0.6 \to k = \displaystyle\frac{\ln(0.6)}{10} \approx -0.051\) Step 5: Solve for the temperature at \(t = 20\) \(T(20) = 20 + 75 e^{20 \cdot \frac{\ln(0.6)}{10}} = 20 + 75(e^{\ln(0.6)})^2 = 20 + 75(0.6)^2\) \(T(20) = 20 + 75(0.36) = 20 + 27 = 47\) \(T(20) = 47^\circ\text{C}\) 5. Homogeneous Second-Order Linear Equations To round out freshman calculus, we often dip our toes into second-order differential equations. The most accessible family is linear, homogeneous equations with constant coefficients. Homogeneous Second-Order Equation and Characteristic Equation These equations have the form: \(a y'' + b y' + c y = 0\) Where \(a\), \(b\), and \(c\) are real constants. We solve them by testing an exponential solution guess, \(y = e^{rx}\). Substituting this guess into the DE yields the quadratic characteristic equation: \(a r^2 + b r + c = 0\) The nature of the roots of this quadratic determines the format of the general solution: Case 1: Two Distinct Real Roots (\(r_1 \neq r_2\)) \(y = C_1 e^{r_1 x} + C_2 e^{r_2 x}\) Case 2: One Repeated Real Root (\(r_1 = r_2 = r\)) \(y = C_1 e^{r x} + C_2 x e^{r x}\) Case 3: Two Complex Roots (\(r = \alpha \pm \beta i\)) \(y = e^{\alpha x} (C_1 \cos(\beta x) + C_2 \sin(\beta x))\) Example 5: Homogeneous Second-Order Initial Value Problem Solve the second-order initial value problem: \(y'' - 5y' + 6y = 0, \quad y(0) = 2, \quad y'(0) = 5\) Step 1: Write the Characteristic Equation \(r^2 - 5r + 6 = 0 \to (r - 2)(r - 3) = 0 \to r_1 = 2, \quad r_2 = 3\) Step 2: Construct the General Solution Since we have two distinct real roots, we use the Case 1 template: \(y(x) = C_1 e^{2x} + C_2 e^{3x}\) Step 3: Use the initial conditions to solve for \(C_1\) and \(C_2\) First, find the derivative \(y'(x)\): \(y'(x) = 2C_1 e^{2x} + 3C_2 e^{3x}\) Now apply our initial conditions at \(x = 0\): \(y(0) = C_1 + C_2 = 2\) \(y'(0) = 2C_1 + 3C_2 = 5\) Solving this basic linear system (multiply the first equation by \(2\) and subtract): \((2C_1 + 3C_2) - (2C_1 + 2C_2) = 5 - 4 \to C_2 = 1\) \(C_1 + 1 = 2 \to C_1 = 1\) Step 4: Write the final particular solution \(y(x) = e^{2x} + e^{3x}\) \(y(x) = e^{2x} + e^{3x}\) Quick-Reference Summary Table Equation Type Standard Structural Form Integration Strategy & Helper Formula First-Order Separable \(\displaystyle\frac{dy}{dx} = g(x)h(y)\) Separate variables: \(\int \displaystyle\frac{1}{h(y)} \, dy = \int g(x) \, dx\) First-Order Linear \(\displaystyle\frac{dy}{dx} + P(x)y = Q(x)\) Integrating Factor: \(I(x) = e^{\int P(x) \, dx} \to y = \displaystyle\frac{1}{I(x)}\int I(x)Q(x) \, dx\) Exponential Model \(\displaystyle\frac{dy}{dt} = ky\) Standard direct solution: \(y(t) = y_0 e^{kt}\) Newton's Law of Cooling \(\displaystyle\frac{dT}{dt} = k(T - T_s)\) Separation yield: \(T(t) = T_s + C e^{kt}\) Second-Order Homogeneous \(a y'' + b y' + c y = 0\) Solve Characteristic Equation: \(a r^2 + b r + c = 0\) Differential equations bridge the gap between algebraic relationships and spatial rates of change. Master standard form alignments, keep your integration properties sharp, and you will unlock the tools to model physical reality itself! See more: Precalculus Complex Numbers Guide: Master Rectangular & Polar Forms First Order Differential Equations LibreTexts: First Order Differential Equations