Worked Problems: Homogeneous DEs with Real, Repeated, and Complex Roots

📚 Homogeneous Differential Equations: A Comprehensive Guide

Homogeneous differential equations are a class of differential equations that can be transformed into a separable form through a suitable substitution. This guide covers homogeneous DEs with real, repeated, and complex roots. Understanding these types of equations is crucial in many areas of physics and engineering.

📜 History and Background

The study of differential equations dates back to the 17th century with the advent of calculus. Mathematicians like Leibniz and Newton laid the groundwork, and the concept of homogeneous equations emerged soon after as a method to simplify and solve more complex problems.

• 🕰️ Early investigations focused on geometric interpretations and physical applications.
• 📈 The formalization of methods for solving homogeneous equations came later, solidifying their place in mathematical analysis.
• 💡 The development of these methods has enabled solutions to various scientific and engineering problems.

🔑 Key Principles

A differential equation is homogeneous if it can be written in the form $\frac{dy}{dx} = f(\frac{y}{x})$. The solution method involves substituting $v = \frac{y}{x}$, which transforms the equation into a separable form. The characteristic equation determines the nature of the roots, which can be real and distinct, real and repeated, or complex conjugates.

➗ Real and Distinct Roots

When the characteristic equation has real and distinct roots $m_1$ and $m_2$, the general solution is given by:

$y(x) = c_1e^{m_1x} + c_2e^{m_2x}$

👯 Real and Repeated Roots

When the characteristic equation has a repeated real root $m$, the general solution is given by:

$y(x) = (c_1 + c_2x)e^{mx}$

complex Complex Roots

When the characteristic equation has complex conjugate roots $\alpha \pm i\beta$, the general solution is given by:

$y(x) = e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))$

⚙️ Worked Problems

🔢 Problem 1: Real and Distinct Roots

Solve the differential equation: $y'' - 3y' + 2y = 0$

Solution:

• ✍️ Write the characteristic equation: $m^2 - 3m + 2 = 0$
• ➗ Factor the equation: $(m - 1)(m - 2) = 0$
• 🌱 Find the roots: $m_1 = 1$, $m_2 = 2$
• ✅ Write the general solution: $y(x) = c_1e^x + c_2e^{2x}$

👯 Problem 2: Real and Repeated Roots

Solve the differential equation: $y'' - 4y' + 4y = 0$

Solution:

• ✍️ Write the characteristic equation: $m^2 - 4m + 4 = 0$
• ➗ Factor the equation: $(m - 2)^2 = 0$
• 🌱 Find the roots: $m = 2$ (repeated)
• ✅ Write the general solution: $y(x) = (c_1 + c_2x)e^{2x}$

🌀 Problem 3: Complex Roots

Solve the differential equation: $y'' + 2y' + 5y = 0$

Solution:

• ✍️ Write the characteristic equation: $m^2 + 2m + 5 = 0$
• ➗ Use the quadratic formula: $m = \frac{-2 \pm \sqrt{2^2 - 4(1)(5)}}{2(1)} = \frac{-2 \pm \sqrt{-16}}{2} = -1 \pm 2i$
• 🌱 Find the roots: $\alpha = -1$, $\beta = 2$
• ✅ Write the general solution: $y(x) = e^{-x}(c_1\cos(2x) + c_2\sin(2x))$

🧪 Problem 4: Initial Value Problem with Complex Roots

Solve the differential equation: $y'' + 4y = 0$, with $y(0) = 0$ and $y'(0) = 2$

Solution:

• ✍️ Characteristic equation: $m^2 + 4 = 0$
• ➗ Roots: $m = \pm 2i$
• 🌱 General solution: $y(x) = c_1 \cos(2x) + c_2 \sin(2x)$
• 🌡️ Apply $y(0) = 0$: $0 = c_1 \cos(0) + c_2 \sin(0) \implies c_1 = 0$
• 📈 $y(x) = c_2 \sin(2x)$, so $y'(x) = 2c_2 \cos(2x)$
• 🔬 Apply $y'(0) = 2$: $2 = 2c_2 \cos(0) \implies c_2 = 1$
• ✅ Final solution: $y(x) = \sin(2x)$

📐 Problem 5: Repeated Roots with Initial Conditions

Solve $y'' - 6y' + 9y = 0$ given $y(0) = 1$ and $y'(0) = 2$

• ✍️ Characteristic equation: $m^2 - 6m + 9 = 0$
• ➗ Factoring: $(m - 3)^2 = 0$, thus $m = 3$ (repeated)
• 🌱 General solution: $y(x) = (c_1 + c_2 x)e^{3x}$
• 🌡️ Applying $y(0) = 1$: $1 = (c_1 + c_2 \cdot 0)e^{0} \implies c_1 = 1$
• 📈 Then, $y'(x) = c_2 e^{3x} + 3(c_1 + c_2 x)e^{3x}$
• 🔬 Applying $y'(0) = 2$: $2 = c_2 e^{0} + 3(1 + c_2 \cdot 0)e^{0} \implies 2 = c_2 + 3 \implies c_2 = -1$
• ✅ Solution: $y(x) = (1 - x)e^{3x}$

📜 Problem 6: Combining Real and Complex Roots

Consider the equation $y''' - y'' + y' - y = 0$. Find the general solution.

• ✍️ Characteristic equation: $m^3 - m^2 + m - 1 = 0$
• ➗ Factoring: $m^2(m - 1) + (m - 1) = 0 \implies (m^2 + 1)(m - 1) = 0$
• 🌱 Roots: $m = 1, \pm i$
• ✅ General Solution: $y(x) = c_1 e^{x} + c_2 \cos(x) + c_3 \sin(x)$

💡 Problem 7: Finding Particular Solution

Suppose you have $(D^2 + 2D + 10)y = 0$ where $D = \frac{d}{dx}$, find its solution.

• ✍️ Characteristic Equation: $m^2 + 2m + 10 = 0$
• ➗ Apply quadratic formula $m = \frac{-2 \pm \sqrt{4 - 40}}{2} = \frac{-2 \pm \sqrt{-36}}{2} = -1 \pm 3i$
• 🌱 Hence complex roots exist: $\alpha = -1, \beta = 3$
• ✅ General solution $y(x) = e^{-x} (c_1\cos 3x + c_2 \sin 3x)$

заключение Conclusion

Understanding homogeneous differential equations and their solutions with various types of roots is essential in mathematics, physics, and engineering. By grasping the concepts and practicing problem-solving, you can master this topic. Keep practicing, and you'll become more comfortable with these equations. 💪