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4.3 Separable Equations

4 min read•june 24, 2024

are a key type of differential equation where variables can be isolated. These equations pop up in many real-world scenarios, from to . They're a great starting point for understanding more complex .

Solving separable equations involves splitting variables, integrating both sides, and solving for y. This process helps us model and predict various phenomena like population growth and . Analyzing solutions through and initial conditions gives us a fuller picture of equation behavior.

Separable Equations

Solving separable differential equations

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  • Separable equations are differential equations that can be written in the form dydx=f(x)g(y)\frac{dy}{dx} = f(x)g(y) where the right-hand side is a product of a function of xx and a function of yy
    • The functions f(x)f(x) and g(y)g(y) depend only on xx and yy, respectively, allowing the variables to be separated
  • To solve a separable equation, first separate the variables by moving all terms involving xx to one side of the equation and all terms involving yy to the other side
    • This step is done by dividing both sides of the equation by g(y)g(y) and multiplying both sides by dxdx
  • After separating the variables, integrate both sides of the equation with respect to their respective variables (xx on the left side and yy on the right side)
    • The integration step often involves techniques such as uu-substitution (change of variables), (sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1), (∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln|x| + C), or
  • Solve the resulting equation for yy as a function of xx, which may require additional steps:
    • Explicitly solve for yy by isolating it on one side of the equation
    • Apply the exponential function to both sides of the equation if the result contains logarithms
    • Determine the using an , which is a known point (x0,y0)(x_0, y_0) that the solution must pass through
    • In some cases, may be necessary to find the solution

Applications of separable equations

  • involve the rate of change of a solution's concentration being proportional to the current concentration
    • The differential equation for this scenario is dCdt=kC\frac{dC}{dt} = kC, where CC is the concentration and kk is a constant (positive for increasing concentration, negative for decreasing)
    • Solve this separable equation to find the concentration as a function of time, which can be used to predict future concentrations or determine the time required to reach a specific concentration
  • Newton's law of cooling states that the rate of change of an object's temperature is proportional to the difference between its temperature and the ambient temperature
    • The differential equation for this scenario is dTdt=k(T−Ta)\frac{dT}{dt} = k(T - T_a), where TT is the object's temperature, TaT_a is the ambient temperature, and kk is a negative constant (cooling rate)
    • Solve this separable equation to find the object's temperature as a function of time, which can be used to predict cooling times or determine the time required to reach a specific temperature
  • Separable equations can also model various other real-world phenomena:
    • Population growth, where the rate of change of a population is proportional to its current size ( or decay)
    • Radioactive decay, where the rate of change of a radioactive substance is proportional to the amount present ()
    • , where the rate of change of an investment's value is proportional to its current value (exponential growth)

Analysis of separable equation solutions

  • Direction fields provide a graphical representation of the slopes of solutions to a differential equation at various points in the xyxy-plane
    • To sketch a direction field, choose several points in the plane, evaluate the differential equation at each point to find the slope, and draw a short line segment with the calculated slope at each point
    • The resulting field gives a visual understanding of the behavior of solutions without explicitly solving the equation, revealing trends, symmetries, and long-term behavior
  • Initial conditions specify a known point (x0,y0)(x_0, y_0) that a solution must pass through, allowing the determination of a from a
    • To find a , solve the separable equation to find the with a constant of integration, then substitute the initial condition and solve for the constant
    • Replace the constant of integration in the general solution with the value found to obtain the particular solution satisfying the initial condition
  • Combining direction fields and initial conditions provides a comprehensive analysis of a separable equation:
    • Sketch the direction field for the given equation to visualize the overall behavior of solutions
    • Plot the initial condition on the direction field to identify the starting point of the particular solution
    • Draw the solution curve passing through the initial condition, following the slopes indicated by the direction field, to trace the path of the particular solution

Autonomous Equations and Equilibrium Solutions

  • are a special type of separable equation where the right-hand side depends only on y (e.g., dydx=f(y)\frac{dy}{dx} = f(y))
  • can be used to visualize the behavior of autonomous equations, with horizontal lines representing constant solutions
  • The provides a one-dimensional representation of the solution behavior for autonomous equations
  • occur when f(y)=0f(y) = 0, representing constant solutions to the differential equation

Key Terms to Review (33)

Autonomous Equations: Autonomous equations are a type of first-order ordinary differential equation (ODE) where the independent variable, usually denoted as 't', does not explicitly appear in the equation. In other words, the equation depends only on the dependent variable and its derivative, and not on the independent variable itself.
Compound interest: Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. It grows at an exponential rate, unlike simple interest which grows linearly.
Compound Interest: Compound interest is the interest earned on interest, where the interest earned in each period is added to the principal, and the total then earns interest in the next period. This concept is fundamental to understanding exponential growth and separable differential equations.
Constant of Integration: The constant of integration is a term that arises in the process of finding the antiderivative or indefinite integral of a function. It represents the unknown value that is added to the result of the integration, ensuring that the final solution satisfies the necessary conditions for the integration process.
Differential Equations: Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are used to model and analyze various phenomena in science, engineering, and other fields where the rate of change of a quantity is of interest.
Direction Fields: A direction field, also known as a slope field or vector field, is a visual representation of the solutions to a first-order differential equation. It provides information about the behavior and characteristics of the solutions without explicitly solving the equation.
Drugs in the bloodstream: Drugs in the bloodstream refer to the concentrations of medicinal compounds that are present within blood plasma. These concentrations can change over time and are often modeled using differential equations.
Equilibrium Solutions: Equilibrium solutions refer to the steady-state solutions of differential equations, where the dependent variable does not change over time. These solutions represent the long-term behavior of a system and are crucial in understanding the dynamics of various physical, biological, and engineering processes.
Exponential decay: Exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. It is commonly modeled with the function $N(t) = N_0 e^{-kt}$, where $N_0$ is the initial quantity, $k$ is the decay constant, and $t$ is time.
Exponential Decay: Exponential decay is a mathematical function that describes the process of a quantity decreasing at a rate proportional to its current value. This term is closely related to the behavior of exponential functions and their applications in various fields, including integrals, logarithms, and the modeling of natural phenomena involving gradual decline or diminishment.
Exponential Growth: Exponential growth is a type of growth pattern where a quantity increases at a rate proportional to its current value. This means that the quantity grows by a consistent percentage over equal intervals of time, leading to a rapid and accelerating increase in its value.
General solution: A general solution to a differential equation includes all possible solutions and typically contains arbitrary constants. It represents the family of curves that satisfies the differential equation.
General Solution: The general solution of a differential equation is the complete set of solutions that satisfies the equation, typically expressed in terms of arbitrary constants that can be adjusted to fit specific initial or boundary conditions.
Half-life: Half-life is the time required for a quantity to reduce to half of its initial value. It is commonly used in contexts involving exponential decay, such as radioactive decay or pharmacokinetics.
Half-life: Half-life is the time it takes for a substance to decay to half of its initial value. It is a fundamental concept in various fields, including radioactive decay, pharmacokinetics, and the study of exponential growth and decay processes.
Implicit Differentiation: Implicit differentiation is a technique used to find the derivative of a function that is not explicitly defined in terms of the independent variable. It involves differentiating both sides of an equation with respect to the independent variable, treating all variables as functions of that variable.
Initial Condition: The initial condition refers to the known value or state of a variable at the starting point of a process or system. It is a crucial piece of information that helps determine the subsequent behavior and solution of a given problem or differential equation.
Logarithmic Properties: Logarithmic properties are the fundamental rules and relationships that govern the behavior of logarithmic functions. These properties describe how logarithms can be manipulated and combined, allowing for efficient calculations and simplification of logarithmic expressions.
Newton’s law of cooling: Newton’s law of cooling states that the rate at which an object changes temperature is proportional to the difference between its own temperature and the ambient temperature. It is commonly expressed using a first-order separable differential equation.
Newton's Law of Cooling: Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between the object's temperature and the temperature of its surroundings. This principle describes the cooling or heating of an object over time and is applicable in various contexts, including exponential growth and decay, separable equations, and first-order linear equations.
Partial Fractions: Partial fractions is a technique used to decompose a rational function into a sum of simpler rational functions. This method is often employed when integrating rational functions, as it allows for the use of inverse trigonometric functions, integration by parts, and other integration techniques.
Particular solution: A particular solution of a differential equation is a specific solution that satisfies both the differential equation and any given initial or boundary conditions. It is distinct from the general solution, which includes a family of solutions with arbitrary constants.
Particular Solution: A particular solution is a specific solution to a differential equation that satisfies the given initial conditions or boundary conditions. It represents one of the solutions to the differential equation that is distinct from the general solution.
Phase Line: A phase line is a graphical representation used in the analysis of separable differential equations. It is a tool that helps visualize the behavior of the solutions to these equations, particularly the direction and stability of the solutions over time.
Radioactive Decay: Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. This spontaneous process is a fundamental concept in the field of nuclear physics and has important applications in various scientific and technological domains.
Separable differential equation: A separable differential equation is a type of ordinary differential equation that can be written as the product of a function of the independent variable and a function of the dependent variable. Such equations can be solved by separating the variables and integrating both sides.
Separable Equations: Separable equations are a type of first-order ordinary differential equation where the variables can be separated, allowing the equation to be solved by integrating each side independently. This concept is central to understanding the fundamental aspects of differential equations and their applications.
Separation of Variables: Separation of variables is a method used to solve ordinary differential equations by breaking down the equation into two or more simpler equations, each involving only one of the variables. This technique allows for the integration of the individual components to obtain the solution to the original differential equation.
Slope Fields: A slope field, also known as a direction field, is a visual representation of the solutions to a first-order ordinary differential equation (ODE). It provides a qualitative understanding of the behavior of the solutions without explicitly solving the equation.
Solution concentrations: Solution concentrations describe the amount of solute dissolved in a solvent, often vital in solving differential equations modeling physical and chemical processes. Concentrations can be expressed in various units such as molarity, molality, or mass percent.
Solution Concentrations: Solution concentration refers to the amount of a solute dissolved in a given volume of solvent, which determines the strength or intensity of the solution. It is an important concept in the study of separable equations, as the concentration of a solution can influence the rate of chemical reactions and the behavior of the system over time.
Trigonometric Identities: Trigonometric identities are mathematical equalities that involve trigonometric functions, such as sine, cosine, and tangent. These identities hold true for any value of the angles involved, and they are fundamental to the study of trigonometry and its applications in various fields, including calculus.
U-substitution: U-substitution is a technique used in integration that simplifies the process by substituting a part of the integral with a new variable, usually denoted as 'u'. This method allows for easier integration by transforming complex expressions into simpler ones, facilitating the calculation of definite and indefinite integrals.
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4.4 The Logistic Equation