Solving Differential Equations: A Comprehensive Guide with an Example

In this article, we will explore the process of solving a specific type of differential equation through substitution. We will delve into the detailed steps, highlighting key mathematical concepts such as substitution, the chain rule, and solving non-homogeneous differential equations. Let's begin with a practical example.

Introduction to Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are a fundamental tool in many areas of science and engineering, including physics, economics, and biology. Solving differential equations involves finding the function(s) that satisfy a given relationship. In this article, we will focus on solving a second-order differential equation involving the exponential function.

The Problem at Hand

Consider the following differential equation:

[ frac{d^2 y}{dx^2} y log t - frac{1}{t^2} ]

where ( t e^x ).

Step 1: Substitution

Given ( t e^x ), we need to express the differential equation in terms of ( t ).

First, let's express ( y ) in terms of ( t ). Define:

[ y yt ]

Now, we need to express the second derivative of ( y ) with respect to ( x ) in terms of ( t ). Using the chain rule:

[ frac{dy}{dx} frac{dt}{dx} frac{dy}{dt} ]

and

[ frac{d^2 y}{dx^2} frac{d}{dx} left( frac{dy}{dx} right) frac{d}{dx} left( frac{dt}{dx} frac{dy}{dt} right) ]

Applying the product rule and chain rule:

[ frac{d^2 y}{dx^2} frac{d^2 t}{dx^2} frac{dy}{dt} frac{dt}{dx}^2 frac{d^2 y}{dt^2} ]

Step 2: Differentiate ( t e^x )

Next, we differentiate ( t e^x ) with respect to ( x ):

[ frac{dt}{dx} e^x t ]

And the second derivative:

[ frac{d^2 t}{dx^2} frac{d}{dx} (t) t ]

Substituting these values into the expression for the second derivative:

[ frac{d^2 y}{dx^2} t frac{dy}{dt} t^2 frac{d^2 y}{dt^2} ]

Step 3: Rewrite the Original Differential Equation

Substitute the expressions for the derivatives into the original differential equation:

[ t^2 frac{d^2 y}{dt^2} t frac{dy}{dt} y log t - frac{1}{t^2} ]

Notice that the equation is homogeneous on the left-hand side:

[ t^2 frac{d^2 y}{dt^2} t frac{dy}{dt} y 0 ]

Thus, the equation simplifies to:

[ t^2 frac{d^2 y}{dt^2} t frac{dy}{dt} y log t - frac{1}{t^2} ]

Step 4: Solving the Non-Homogeneous Equation

To solve this non-homogeneous equation, we need to find a particular solution to the non-homogeneous part and then combine it with the solution to the homogeneous equation.

Homogeneous Equation

The homogeneous part of the equation is:

[ t^2 frac{d^2 y}{dt^2} t frac{dy}{dt} y 0 ]

This is a second-order linear differential equation. The characteristic equation is:

[ t^2 r^2 t r 1 0 ]

Solving for ( r ) using the quadratic formula:

[ r frac{-1 pm sqrt{1 - 4}}{2t} frac{-1 pm sqrt{-3}}{2t} ]

This gives complex roots:

[ r frac{-1 pm isqrt{3}}{2t} ]

Therefore, the complementary solution is:

[ y_c t^{-1} left( A cos left( frac{sqrt{3}}{2} ln t right) B sin left( frac{sqrt{3}}{2} ln t right) right) ]

Particular Solution

To find a particular solution, we can use methods such as variation of parameters or undetermined coefficients. Here, we will use undetermined coefficients. Assume a particular solution of the form:

[ y_p frac{At B}{t^2} C log t D frac{1}{t^2} ]

Substitute ( y_p ) into the non-homogeneous equation and solve for the constants ( A ), ( B ), ( C ), and ( D ).

Step 5: General Solution

The general solution is the sum of the complementary solution and the particular solution:

[ y y_c y_p ]

Substitute the values of the constants obtained in the previous step to get the final solution.

Conclusion

In this article, we have explored the process of solving a second-order differential equation using substitution and the chain rule. The solution involves recognizing the homogeneous and non-homogeneous parts of the equation and finding both the complementary and particular solutions.

Key Concepts

Differential Equations: Equations involving derivatives that describe a relationship between a function and its derivatives. Substitution: A technique to transform a given equation into a more manageable form. Chain Rule: A rule in calculus for differentiating composite functions. Non-Homogeneous Equations: Equations that contain a non-zero term that does not depend on the dependent variable.

Further Reading

For further exploration, consider studying the following topics:

General methods for solving differential equations. Advanced techniques in solving non-homogeneous differential equations. Applications of differential equations in real-world scenarios.

Questions and Answers

If you have any questions or need clarification on any part of the solution, feel free to ask in the comments below. We will be happy to help!