Further Differential Equations and Modelling

Mastering differential equations is the key to moving from simply describing the world to predicting and controlling it. This article equips you with advanced techniques to solve complex equations and apply them to realistic models, from vibrating bridges to interacting animal populations. You will learn to handle equations where the coefficients are not constant, analyze interconnected systems, and extract meaningful physical insight from your mathematical solutions.

Solving Second-Order Equations with Variable Coefficients

Many physical laws lead to second-order differential equations where the coefficients are functions of the independent variable, not constants. A standard linear second-order equation has the form $a(x)\frac{d^{2}y}{d{x}^2}+b(x)\frac{dy}{dx}+c(x)y=f(x)$. When $a(x)$, $b(x)$, and $c(x)$ are not all constants, the characteristic equation method fails, and we must turn to substitution techniques.

The most common strategy involves a change of independent variable to transform the equation into one with constant coefficients, which we can then solve. A prime candidate for this method is the Euler-Cauchy equation, which has the form $a{x}^2\frac{d^{2}y}{d{x}^2}+bx\frac{dy}{dx}+cy=0$, where $a$, $b$, and $c$ are constants. The substitution $x=e^t$ (and thus $t=\ln x$) is used. This leverages the chain rule: $\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}=\frac{1}{x}\frac{dy}{dt}$. The second derivative becomes $\frac{d^{2}y}{d{x}^2}=\frac{1}{x^2}\left(\frac{d^{2}y}{d{t}^2}-\frac{dy}{dt}\right)$. Substituting these into the original equation transforms it into a standard linear equation with constant coefficients in terms of $t$: $a\frac{d^{2}y}{d{t}^2}+(b-a)\frac{dy}{dt}+cy=0$.

Consider solving $x^2{y}^{\prime \prime }-2x{y}^{\prime }+2y=0$. Using the substitution $x=e^t$, we get $\frac{d^{2}y}{d{t}^2}-3\frac{dy}{dt}+2y=0$. The characteristic equation is $m^2-3m+2=0$, with roots $m=1$ and $m=2$. Therefore, the general solution in $t$ is $y=A{e}^t+B{e}^{2t}$. Re-substituting $t=\ln x$ gives the final solution in terms of $x$: $y=Ax+B{x}^2$. This method is powerful because it converts an unfamiliar problem into a familiar one, allowing you to apply all the techniques you know for constant coefficient equations, including handling inhomogeneous terms.

Systems of Coupled First-Order Equations

Real-world systems often involve multiple interdependent quantities changing simultaneously. These are modelled by systems of coupled first order differential equations. A standard linear system with two dependent variables $x(t)$ and $y(t)$ looks like: $$\frac{dx}{dt}=a_{11}x+a_{12}y$$ $$\frac{dy}{dt}=a_{21}x+a_{22}y$$ The key to solving such a system is to recognize it can be written elegantly in matrix form: $$\frac{d}{dt}\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$$ or simply $\frac{d\mathbf{x}}{dt}=A\mathbf{x}$, where $\mathbf{x}$ is the state vector and $A$ is the coefficient matrix.

The solution hinges on eigenvalue analysis. We look for solutions of the form $\mathbf{x}=\mathbf{v}{e}^{\lambda t}$, where $\mathbf{v}$ is a constant vector. Substituting into the matrix equation gives $A\mathbf{v}=\lambda \mathbf{v}$. This is the classic eigenvalue problem. The values of $\lambda$ are found by solving the characteristic equation $\det (A-\lambda I)=0$. For each distinct real eigenvalue ${\lambda }_i$, we find its corresponding eigenvector ${\mathbf{v}}_i$. The general solution is then a linear combination: $\mathbf{x}(t)=C_1{\mathbf{v}}_1{e}^{{\lambda }_{1}t}+C_2{\mathbf{v}}_2{e}^{{\lambda }_{2}t}$. If the eigenvalues are complex conjugates ($\alpha \pm i\beta$), the solution will involve sines and cosines, indicating oscillatory behavior. This matrix-eigenvalue method uncouples the system, providing a direct path to the solution and a clear window into the system's dynamics.

Modelling and Interpreting Physical Systems

The true power of these techniques is revealed when we apply them to model tangible phenomena. The solutions are not just abstract functions; they describe observable behavior, stability, and long-term trends.

Damped and Forced Oscillations: The motion of a mass on a spring, an electrical charge in an LC circuit, or a swaying building is modelled by a second-order constant coefficient ODE: $m\ddot{x}+c\dot{x}+kx=F(t)$. Here, $m$ is mass/inertia, $c$ is the damping coefficient, $k$ is the stiffness, and $F(t)$ is an external force. The homogeneous solution ($F(t)=0$) describes the system's natural response: undamped oscillation ($c=0$), underdamped decay (complex roots), critically damped return (repeated real root), or overdamped return (distinct real roots). Adding a periodic forcing term, like $F_0\cos (\omega t)$, leads to a particular solution representing a steady-state oscillation. When the forcing frequency $\omega$ nears the system's natural frequency, resonance occurs, leading to large-amplitude oscillations—a critical consideration in engineering design to prevent structural failure.

Predator-Prey Dynamics: The classic Lotka-Volterra equations model the interaction between two species: $$\frac{dR}{dt}=\alpha R-\beta RF$$ $$\frac{dF}{dt}=-\gamma F+\delta RF$$ Here, $R(t)$ is the prey population, $F(t)$ is the predator population, and $\alpha ,\beta ,\gamma ,\delta$ are positive constants. This is a nonlinear coupled system. We analyze its long-term behaviour by finding equilibrium points (where both derivatives are zero) and examining stability. The non-zero equilibrium is at $(R,F)=(\gamma /\delta ,\alpha /\beta )$. Linearising the system around this point (using the Jacobian matrix) and performing eigenvalue analysis reveals purely imaginary eigenvalues. This indicates closed orbits in the phase plane, predicting cyclical population booms and busts for both species, a key insight into ecosystem stability.

Electrical Circuits: An RLC circuit (Resistor, Inductor, Capacitor in series) is governed by an equation perfectly analogous to the mechanical oscillator: $L\frac{d^{2}q}{d{t}^2}+R\frac{dq}{dt}+\frac{1}{C}q=E(t)$, where $q$ is charge, $L$ is inductance, $R$ is resistance, $C$ is capacitance, and $E(t)$ is the applied voltage. The interpretation is direct: inductance $L$ provides inertia against current change (like mass), resistance $R$ dissipates energy as heat (like damping), and capacitance $C$ stores potential energy (like the spring). Solving this equation tells you the transient and steady-state current or voltage anywhere in the circuit, which is fundamental to designing filters, tuners, and signal processors.

Common Pitfalls

1. Misapplying the Characteristic Equation Method: The most common error is trying to use the characteristic equation $a{\lambda }^2+b\lambda +c=0$ for an equation like $x^2{y}^{\prime \prime }+x{y}^{\prime }=0$. This only works for constant coefficients $a,b,c$. For variable coefficients, you must use a substitution method like the one for Euler-Cauchy equations.

• Correction: First identify the type of equation. If it is an Euler-Cauchy form, use the $x=e^t$ substitution. For other variable coefficient equations, you may need more advanced techniques like reduction of order or series solutions.

1. Confusing Matrix Operations in System Solutions: When solving $\frac{d\mathbf{x}}{dt}=A\mathbf{x}$, a frequent mistake is to incorrectly handle the eigenvectors when eigenvalues are complex or repeated. Another error is forgetting that the arbitrary constants $C_1$ and $C_2$ multiply the entire solution vector for each mode.

• Correction: For complex eigenvalues $\lambda =\alpha \pm i\beta$, derive two linearly independent real solutions from the real and imaginary parts of $\mathbf{v}{e}^{\lambda t}$. Always write the final solution as $\mathbf{x}(t)=C_1(\text{first solution vector})+C_2(\text{second solution vector})$.

1. Neglecting the Physical Interpretation of Parameters and Solutions: It's easy to get lost in the algebra and forget what the solution represents. For example, finding a general solution to a damped oscillation equation without identifying whether the system is underdamped or overdamped misses the core physical insight.

• Correction: Always relate mathematical results back to the model's parameters. After solving, state explicitly: "Because the discriminant $c^2-4mk$ is negative, the system is underdamped and will oscillate as it decays." In population models, explicitly find and interpret equilibrium points and their stability.

1. Incorrectly Linearising Nonlinear Systems: When analyzing stability for systems like predator-prey models, a mistake is to use the original nonlinear terms in the Jacobian matrix or to miscalculate the partial derivatives at the equilibrium point.

• Correction: To linearise a system $\frac{dx}{dt}=f(x,y),\frac{dy}{dt}=g(x,y)$ at point $(x_0,y_0)$, construct the Jacobian matrix $J=\left(\begin{array}{cc}{f}_x& f_y\\ g_x& g_y\end{array}\right)$ evaluated at $(x_0,y_0)$. The eigenvalues of this matrix $J$ determine the local stability of the equilibrium in the original nonlinear system.

Summary

• Second-order ODEs with variable coefficients, such as the Euler-Cauchy equation, are solved using strategic substitutions (like $x=e^t$) to transform them into constant-coefficient equations that can be solved via their characteristic equations.
• Systems of coupled linear first-order ODEs are efficiently solved by rewriting them in matrix form $\frac{d\mathbf{x}}{dt}=A\mathbf{x}$ and finding the eigenvalues and eigenvectors of the coefficient matrix $A$. The general solution is a combination of terms like $\mathbf{v}{e}^{\lambda t}$.
• Mathematical models like the damped/forced harmonic oscillator, Lotka-Volterra predator-prey equations, and RLC circuit equations translate physical laws into solvable differential equations. The parameters in these equations have direct physical meanings (mass, damping, growth rate, capacitance, etc.).
• Analysis of solutions involves interpreting the mathematical output (exponential decay, oscillation, equilibrium values) in the context of the original problem. This includes determining long-term behavior (steady-state, cycles, extinction) and stability, often using phase plane analysis and linearisation for nonlinear systems.
• Eigenvalue analysis is the unifying powerful tool, determining the nature of solutions for both single higher-order equations (through the characteristic equation) and systems of equations (through the matrix eigenvalue problem), directly revealing whether a system oscillates, grows, or decays.