Solving Non-Homogeneous Differential Equations: The Method of Undetermined Coefficients

Understanding how to solve non-homogeneous differential equations, such as the second-order linear equation provided, is crucial in many fields of science and engineering. This article will guide you through the process of solving a specific differential equation using the method of undetermined coefficients.

Introduction to Differential Equations

Differential equations are fundamental in modeling a wide range of phenomena in physics, engineering, and other sciences. They describe how a quantity changes over time or space. In this article, we focus on non-homogeneous differential equations of the form:

frac{d^2y}{dx^2} - 5frac{dy}{dx} - 4y 10e^{-3x}

Here, the term 10e^{-3x} is the non-homogeneous part, meaning that the equation involves an external input or forcing function.

Step-by-Step Solution

Step 1: Solve the Homogeneous Equation

The first step in solving the non-homogeneous equation is to solve the associated homogeneous equation:

frac{d^2y}{dx^2} - 5frac{dy}{dx} - 4y 0

To solve this, we assume a solution of the form:

y e^{rx}

Substituting this into the homogeneous equation, we get the characteristic equation:

r^2 - 5r - 4 0

Solving the characteristic equation using the quadratic formula, we find:

r frac{-(-5) pm sqrt{(-5)^2 - 4(1)(-4)}}{2(1)} frac{5 pm sqrt{25 16}}{2} frac{5 pm 9}{2}

This gives us the roots:

r_1 -1 r_2 -4

Thus, the complementary solution is:

y_c C_1 e^{-x} C_2 e^{-4x}

Step 2: Find a Particular Solution

Next, we need to find a particular solution of the form:

y_p Ae^{-3x}

where A is a constant to be determined. Calculating the first and second derivatives of y_p:

frac{dy_p}{dx} -3Ae^{-3x} frac{d^2y_p}{dx^2} 9Ae^{-3x}

Substituting these into the original non-homogeneous equation:

9Ae^{-3x} - 5(-3Ae^{-3x}) - 4Ae^{-3x} 10e^{-3x}

Simplifying, we get:

9A - 15A 4Ae^{-3x} 10e^{-3x}

-2Ae^{-3x} 10e^{-3x}

Setting the coefficients equal, we find:

-2A 10 implies A -5

Thus, the particular solution is:

y_p -5e^{-3x}

Step 3: Write the General Solution

The general solution y to the differential equation is the sum of the complementary and particular solutions:

y y_c y_p C_1 e^{-x} C_2 e^{-4x} - 5e^{-3x}

Final Solution

The solution to the differential equation frac{d^2y}{dx^2} - 5frac{dy}{dx} - 4y 10e^{-3x} is:

y C_1 e^{-x} C_2 e^{-4x} - 5e^{-3x}

where C_1 and C_2 are constants determined by initial or boundary conditions.

Conclusion

Understanding the method of undetermined coefficients is crucial for solving non-homogeneous differential equations. This process involves finding the complementary solution to the homogeneous equation and a particular solution to the non-homogeneous equation, and then combining them to form the general solution.

Recommended Reading

Nonhomogeneous Linear Systems Undetermined Coefficients Formula