9.3 Differential Equation Solving

Differential equations are powerful tools for modeling real-world phenomena. They come in various types, classified by order, linearity, and homogeneity. Understanding these classifications is crucial for selecting the appropriate solution method.

Solving differential equations involves techniques like separation of variables and integrating factors for first-order equations. For more complex cases, methods like undetermined coefficients and variation of parameters are used. These techniques enable us to tackle a wide range of practical problems.

Classification and Solution Methods for Differential Equations

Classification of differential equations

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• Order of a differential equation determined by the highest derivative present
  • First-order differential equation has the first derivative as the highest derivative (e.g., $\frac{dy}{dx} = x^2 + y$)
  • Second-order differential equation has the second derivative as the highest derivative (e.g., $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 2y = 0$)
• Linearity of a differential equation depends on how the dependent variable and its derivatives appear
  • Linear differential equation has the dependent variable and its derivatives appearing linearly with coefficients being functions of the independent variable only (e.g., $\frac{dy}{dx} + 2xy = 0$)
  • Non-linear differential equation has the dependent variable or its derivatives appearing non-linearly (e.g., $\frac{dy}{dx} = y^2 + x$)
• Homogeneity of a differential equation determined by the right-hand side of the equation
  • Homogeneous differential equation has a right-hand side equal to zero (e.g., $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 2y = 0$)
  • Non-homogeneous differential equation has a non-zero function on the right-hand side (e.g., $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 2y = e^x$)

Solving first-order linear equations

• Separation of variables method applies to first-order differential equations of the form $\frac{dy}{dx} = f(x)g(y)$
  1. Separate variables by moving terms with $y$ to one side and terms with $x$ to the other side
  2. Integrate both sides of the equation
  3. Solve for the dependent variable
• Integrating factor method applies to first-order linear differential equations of the form $\frac{dy}{dx} + P(x)y = Q(x)$
  1. Find the integrating factor $\mu(x) = e^{\int P(x) dx}$
  2. Multiply both sides of the equation by the integrating factor
  3. Simplify the left-hand side to obtain $\frac{d}{dx}(\mu(x)y)$
  4. Integrate both sides of the equation
  5. Solve for the dependent variable

Advanced Solution Methods and Applications

Methods for non-homogeneous equations

• Method of undetermined coefficients applies to non-homogeneous linear differential equations with specific right-hand side functions (polynomials, exponentials, sines, or cosines)
  1. Find the general solution to the corresponding homogeneous equation
  2. Assume a particular solution based on the form of the right-hand side function
  3. Substitute the assumed solution into the differential equation and solve for the unknown coefficients
  4. Add the particular solution to the general solution of the homogeneous equation
• Variation of parameters method applies to non-homogeneous linear differential equations with any right-hand side function
  1. Find the general solution to the corresponding homogeneous equation
  2. Assume a particular solution as a linear combination of the fundamental solutions with variable coefficients
  3. Substitute the assumed solution into the differential equation and solve for the variable coefficients
  4. Integrate the variable coefficients and substitute the results into the assumed particular solution
  5. Add the particular solution to the general solution of the homogeneous equation

Systems of linear differential equations

• Eigenvalues and eigenvectors
  • Eigenvalue $\lambda$ is a scalar that satisfies the equation $A\vec{v} = \lambda\vec{v}$ for a square matrix $A$
  • Eigenvector $\vec{v}$ is a non-zero vector that satisfies the equation $A\vec{v} = \lambda\vec{v}$ for a square matrix $A$
• Solving systems of linear differential equations
  1. Convert the system of differential equations into a matrix form $\frac{d\vec{x}}{dt} = A\vec{x}$
  2. Find the eigenvalues and eigenvectors of the coefficient matrix $A$
  3. Express the general solution as a linear combination of the eigenvectors multiplied by exponential functions of the eigenvalues

Applications in real-world modeling

• Population growth models
  • Exponential growth: $\frac{dP}{dt} = kP$, where $P$ is the population and $k$ is the growth rate
  • Logistic growth: $\frac{dP}{dt} = kP(1 - \frac{P}{K})$, where $K$ is the carrying capacity
• Radioactive decay: $\frac{dN}{dt} = -\lambda N$, where $N$ is the number of atoms and $\lambda$ is the decay constant
• Mechanical vibrations: $m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F(t)$, where $m$ is mass, $c$ is damping coefficient, $k$ is spring constant, and $F(t)$ is the external force
• Electrical circuits: $L\frac{dI}{dt} + RI + \frac{1}{C}\int I dt = V(t)$, where $L$ is inductance, $R$ is resistance, $C$ is capacitance, $I$ is current, and $V(t)$ is the applied voltage