Solving (1+x)^2 D^2y/dx^2 + (1+x) Dy/dx + Y = Cos(log(1+x)) A Step-by-Step Guide

Introduction

In this article, we will delve into the process of solving a second-order linear differential equation with variable coefficients. Specifically, we will tackle the equation:

(1+x)^2 d^2y/dx2 + (1+x) dy/dx + y = cos(log(1+x))

This type of equation often arises in various fields of engineering and physics, making its solution a crucial skill for students and professionals alike. We will employ a strategic approach involving a suitable substitution to transform the given equation into a more manageable form – a linear differential equation with constant coefficients. By solving this transformed equation, we can then revert to the original variable to obtain the general solution of the initial problem. This article will provide a step-by-step guide, ensuring clarity and understanding at each stage of the solution process.

1. Transforming the Equation: A Strategic Substitution

The presence of the terms (1+x)^2 and (1+x) as coefficients of the second and first-order derivatives, respectively, suggests a particular substitution to simplify the equation. Our key strategy here is to introduce a new independent variable, z, defined by:

z = log(1+x)

This substitution is motivated by the fact that the derivatives of (1+x) will neatly cancel out with the terms in the coefficients. The goal is to transform the given differential equation into an equivalent one with constant coefficients, which are significantly easier to solve. To implement this substitution effectively, we need to express the derivatives dy/dx and d^2y/dx2 in terms of derivatives with respect to z.

From the substitution, we can deduce the relationship between x and z as:

1 + x = e^z

Now, let's find the first derivative, dy/dx, using the chain rule:

dy/dx = (dy/dz) * (dz/dx)

We know that z = log(1+x), so dz/dx = 1/(1+x). Substituting this back into the equation, we get:

dy/dx = (dy/dz) * (1/(1+x)) = (dy/dz) * e^(-z)

Next, we find the second derivative, d^2y/dx2, by differentiating dy/dx with respect to x again. This requires the product rule and the chain rule:

d^2y/dx2 = d/dx [(dy/dz) * e^(-z)]

= [d^2y/dz2 * (dz/dx) * e^(-z)] + [dy/dz * (-e^(-z)) * (dz/dx)]

Substituting dz/dx = 1/(1+x) = e^(-z), we have:

d^2y/dx2 = [d^2y/dz2 * e^(-z) * e^(-z)] + [dy/dz * (-e^(-z)) * e^(-z)]

= e^(-2z) * (d^2y/dz2 - dy/dz)

With these expressions for dy/dx and d^2y/dx2 in terms of z, we can now substitute them into the original differential equation. This crucial step will transform the equation into a form that we can solve more readily.

2. Substituting and Simplifying: Constant Coefficients Emerge

Now that we have the expressions for dy/dx and d^2y/dx2 in terms of z, the next crucial step is to substitute these into the original differential equation:

(1+x)^2 d^2y/dx2 + (1+x) dy/dx + y = cos(log(1+x))

Recall that we have the following substitutions:

• 1 + x = e^z
• dy/dx = (dy/dz) * e^(-z)
• d^2y/dx2 = e^(-2z) * (d^2y/dz2 - dy/dz)

Substituting these into the equation, we get:

(e^z)2 * [e^(-2z) * (d^2y/dz2 - dy/dz)] + (e^z) * [(dy/dz) * e^(-z)] + y = cos(z)

Now, we simplify the equation. Notice how the exponential terms neatly cancel out, leading to a significant simplification:

(e^(2z) * e^(-2z) * (d^2y/dz2 - dy/dz)) + (e^z * e^(-z) * (dy/dz)) + y = cos(z)

This simplifies to:

(d^2y/dz2 - dy/dz) + (dy/dz) + y = cos(z)

Observe that the terms involving dy/dz cancel each other out, leading to a remarkably simpler equation:

d^2y/dz2 + y = cos(z)

This is a second-order linear differential equation with constant coefficients! This transformation is the key to solving the original problem. Equations with constant coefficients have well-established solution methods, making this simplified form significantly easier to handle. We have successfully transformed a complex equation with variable coefficients into a standard form that we can readily solve.

3. Solving the Transformed Equation: Finding the General Solution

We've successfully transformed the original differential equation into a simpler form with constant coefficients:

d^2y/dz2 + y = cos(z)

To solve this equation, we need to find the general solution, which consists of two parts: the complementary function (yc) and the particular integral (yp). The complementary function is the general solution of the homogeneous equation (the equation with the right-hand side set to zero), and the particular integral is a specific solution that satisfies the non-homogeneous equation.

3.1. Finding the Complementary Function (yc)

To find the complementary function, we consider the homogeneous equation:

d^2y/dz2 + y = 0

We assume a solution of the form y = e^(mz), where m is a constant. Substituting this into the homogeneous equation gives the auxiliary equation:

m^2 * e^(mz) + e^(mz) = 0

Dividing by e^(mz) (since it's never zero), we get the auxiliary equation:

m^2 + 1 = 0

Solving for m, we find:

m^2 = -1

m = ±i

Since the roots are complex conjugates (m = ±i), the complementary function is of the form:

yc = A * cos(z) + B * sin(z)

where A and B are arbitrary constants. This represents the general solution to the homogeneous equation.

3.2. Finding the Particular Integral (yp)

Now, we need to find a particular integral that satisfies the non-homogeneous equation:

d^2y/dz2 + y = cos(z)

Since the right-hand side is cos(z), we might initially assume a particular integral of the form yp = C * cos(z), where C is a constant. However, we notice that cos(z) is already part of the complementary function. This means we need to modify our assumed form to avoid redundancy. A common technique in such cases is to multiply the initial guess by z.

So, we assume a particular integral of the form:

yp = Cz * sin(z) + Dz * cos(z)

where C and D are constants to be determined. We need to find the first and second derivatives of yp:

dy_p/dz = C * sin(z) + Cz * cos(z) + D * cos(z) - Dz * sin(z)

d^2y_p/dz2 = C * cos(z) + C * cos(z) - Cz * sin(z) - D * sin(z) - D * sin(z) - Dz * cos(z)

= 2C * cos(z) - Cz * sin(z) - 2D * sin(z) - Dz * cos(z)

Now, we substitute yp and d^2yp/dz2 into the non-homogeneous equation:

(2C * cos(z) - Cz * sin(z) - 2D * sin(z) - Dz * cos(z)) + (Cz * sin(z) + Dz * cos(z)) = cos(z)

Simplifying, we get:

2C * cos(z) - 2D * sin(z) = cos(z)

By comparing coefficients of cos(z) and sin(z), we can determine the values of C and D:

• 2C = 1 => C = 1/2
• -2D = 0 => D = 0

Therefore, the particular integral is:

yp = (1/2)z * sin(z)

3.3. The General Solution

The general solution to the differential equation is the sum of the complementary function and the particular integral:

y(z) = yc + yp

y(z) = A * cos(z) + B * sin(z) + (1/2)z * sin(z)

where A and B are arbitrary constants. We have now found the general solution in terms of the variable z. The final step is to transform this solution back into terms of the original variable x.

4. Transforming Back to the Original Variable: The Final Solution

We have obtained the general solution in terms of z:

y(z) = A * cos(z) + B * sin(z) + (1/2)z * sin(z)

To express the solution in terms of the original variable x, we need to substitute back z = log(1+x):

y(x) = A * cos(log(1+x)) + B * sin(log(1+x)) + (1/2) * log(1+x) * sin(log(1+x))

This is the general solution of the given differential equation in terms of x. It represents a family of solutions, parameterized by the arbitrary constants A and B. These constants can be determined if initial conditions or boundary conditions are provided.

Conclusion

In this article, we have successfully solved the second-order linear differential equation with variable coefficients:

(1+x)^2 d^2y/dx2 + (1+x) dy/dx + y = cos(log(1+x))

The key to solving this equation was the strategic substitution z = log(1+x), which transformed the equation into a simpler form with constant coefficients. We then found the complementary function and particular integral of the transformed equation and combined them to obtain the general solution in terms of z. Finally, we substituted back to express the solution in terms of the original variable x.

This process demonstrates a powerful technique for solving differential equations with variable coefficients: transforming them into more manageable forms using appropriate substitutions. The general solution we obtained, y(x) = A * cos(log(1+x)) + B * sin(log(1+x)) + (1/2) * log(1+x) * sin(log(1+x)), provides a complete description of the solutions to the given differential equation. This method is applicable to a wide range of similar problems, making it a valuable tool in mathematical analysis and its applications.