Problem 15 - Differential Equations and Linear Algebra

Question

Solve the given differential equation on the interval \(x>0 .\) [Remember to put the equation in standard form.] $$x^{2} y^{\prime \prime}+4 x y^{\prime}+2 y=\cos x$$

Step-by-Step Solution

Answer

The general solution of the given second-order linear non-homogeneous differential equation with variable coefficients \[y'' + \frac{4}{x}y' + \frac{2}{x^2}y = \frac{\cos x}{x^2}\] can be found using the method of Variation of Parameters. First, find the complementary functions (\(y_1(x)\) and \(y_2(x)\)) of the associated homogeneous equation. Then, construct the particular solution \(y_p(x) = u(x)y_1(x) + v(x)y_2(x)\), where \(u(x)\) and \(v(x)\) are obtained by solving a system of linear equations involving their derivatives. Finally, the general solution is given by \(y(x) = c_1y_1(x) + c_2y_2(x) + y_p(x)\), where \(c_1\) and \(c_2\) are arbitrary constants.

Step 1: Rewrite the differential equation in the standard form

Divide the entire differential equation by \(x^2\): \[\frac{1}{x^2}\left(x^2 y'' + 4xy' + 2y\right) = \frac{1}{x^2}\left(\cos x\right)\] This simplifies to: \[y'' + \frac{4}{x}y' + \frac{2}{x^2}y = \frac{\cos x}{x^2}\] Now the differential equation is in the standard form.

Step 2: Identify the type of differential equation

We can see that the given differential equation is a second-order linear non-homogeneous differential equation with variable coefficients.

Step 3: Use an appropriate solving method

For this type of differential equation, we can use the method of Variation of Parameters. We first need to find the complementary functions, which are the solutions of the associated homogeneous equation: \[y'' + \frac{4}{x}y' + \frac{2}{x^2}y = 0\]

Step 4: Find the complementary functions

The complementary functions can be found either by factoring the associated homogeneous equation or by using other methods such as Frobenius method or power series method. For simplicity, let's assume that we have already found the two linearly independent solutions, denoted by \(y_1(x)\) and \(y_2(x)\).

Step 5: Apply the method of Variation of Parameters

Let \(y_p(x) = u(x)y_1(x) + v(x)y_2(x)\) be the particular solution of the given non-homogeneous differential equation. Apply the method of Variation of Parameters to find the functions \(u(x)\) and \(v(x)\), where \(u'(x)y_1(x) + v'(x)y_2(x) = 0\), and \(u'(x)y_1'(x) + v'(x)y_2'(x) = \frac{\cos x}{x^2}\).

Step 6: Solve for \(u'(x)\) and \(v'(x)\) (integrating)

Now we have a system of two linear equations with two unknowns, \(u'(x)\) and \(v'(x)\). Solve this system to find \(u'(x)\) and \(v'(x)\). Integrate each of them to find the functions \(u(x)\) and \(v(x)\).

Step 7: Construct the particular solution

With the functions \(u(x)\) and \(v(x)\) obtained in the previous step, construct the particular solution \(y_p(x) = u(x)y_1(x) + v(x)y_2(x)\).

Step 8: Write the general solution

The general solution of the given non-homogeneous differential equation is the sum of the complementary functions and the particular solution, i.e.: \[y(x) = c_1y_1(x) + c_2y_2(x) + y_p(x)\] Here, \(c_1\) and \(c_2\) represent arbitrary constants.

Key Concepts

Second-order Linear Differential Equation

A second-order linear differential equation is a type of equation that involves an unknown function, its first derivative, and its second derivative. These equations appear in the form:\[ a(x) y'' + b(x) y' + c(x) y = g(x) \]where:

• \(y''\) is the second derivative of an unknown function \(y\).
• \(y'\) is the first derivative of \(y\).
• \(a(x), b(x), c(x)\) are functions of the independent variable, typically \(x\).
• \(g(x)\) is a given function, representing the non-homogeneous part.

Here, the focus is on ensuring the equation is linear, meaning that there are no products or powers of the function \(y\), \(y'\), \(y''\). This standard form allows mathematicians to apply specific methods to find the solutions, depending on whether it's a homogeneous equation or a non-homogeneous one.

Homogeneous Equation

A homogeneous differential equation is a special type of second-order linear equation where the function on the right-hand side, \(g(x)\), is equal to zero:\[ a(x) y'' + b(x) y' + c(x) y = 0 \]This indicates that the solution to the equation will rely entirely on the properties of the differential equation itself without being influenced by any external terms. The process of solving involves finding the complementary function, denoted as \(y_c(x)\). All solutions to this form of equation are solutions to the corresponding non-homogeneous equation.
The complementary function consists of a linear combination of linearly independent solutions, usually written as \(c_1 y_1(x) + c_2 y_2(x)\), where \(c_1\) and \(c_2\) are constants. These constants come into play when applying initial or boundary conditions to the problem. Often, for relatively simple homogeneous equations, solutions can be found using methods like factoring, or series solutions if the equation has variable coefficients.

Variation of Parameters

Variation of Parameters is a method used to find particular solutions of a non-homogeneous differential equation. This method is especially useful when the associated homogeneous equation is already solved. Assuming you have a general solution for the homogeneous equation:\[ y_c(x) = c_1 y_1(x) + c_2 y_2(x) \]You'll look to find a particular solution \(y_p(x)\) of the form:\[ y_p(x) = u(x) y_1(x) + v(x) y_2(x) \]Here \(u(x)\) and \(v(x)\) are functions determined by 'varying' the constants in the complementary function solution to compensate for the non-homogeneous part \(g(x)\).
To determine \(u(x)\) and \(v(x)\), solve the system:

• \(u'(x)y_1(x) + v'(x)y_2(x) = 0\)
• \(u'(x)y_1'(x) + v'(x)y_2'(x) = \frac{g(x)}{a(x)}\)

Solving this system through integration will provide the functions \(u(x)\) and \(v(x)\), allowing you to construct the particular solution \(y_p(x)\). This technique expands the solution from just relying on the roots of the characteristic equation, offering a powerful approach to solving more complex differential equations.

Non-homogeneous Differential Equation

A non-homogeneous differential equation includes an external function \(g(x)\), making the right side of the equation non-zero:\[ a(x) y'' + b(x) y' + c(x) y = g(x) \]The presence of \(g(x)\) necessitates the search for a specific solution in addition to the general solution of the associated homogeneous equation.Unlike homogeneous equations, these require considering both complementary and particular solutions to form the general solution:\[ y(x) = y_c(x) + y_p(x) \]where \(y_c(x)\) is the complementary solution derived from the homogeneous part, and \(y_p(x)\) is a particular solution catering to \(g(x)\).
There are several methods to find the particular solution, but Variation of Parameters and the Method of Undetermined Coefficients are the most common. Variation of Parameters is preferred for complex \(g(x)\) expressions or when the coefficients of the equation are variable, providing a systematic approach to determine \(y_p(x)\). This makes these equations a bit more intricate than their homogeneous counterparts, but also notably useful in practical applications like engineering and physics, where external forces or influences are prevalent.