Question

Solve the following differential equations by power series and also by an elementary method. Verify that the series solution is the power series expansion of your other solution. $$ y^{\prime \prime}+y=4 \sin 3 x $$

Short answer

The solution is -In$t -3x-33 This is the necessary step.

Step-by-step solution

- Power Series Assumption

Substitute these back into the differential equation: Y{n=2 Putting these together, we need to solve the recurrence relation for the coefficients. Using the method of undetermined coefficients, we find the nons n = Applied, Special \(\text{ special methods and concepts like equations until we get a simplified series expression}^\)

Key concepts

Power Series Solution

The power series method is a powerful tool for solving differential equations. The basic idea is to assume the solution can be written as an infinite sum, or power series. For example, consider the differential equation \( y'' + y = 4 \sin 3x \). To solve this using the power series method, we assume the solution can be written as:
\[ y(x) = \sum_{n=0}^{fty} a_n x^n \]
where \( a_n \) are coefficients to be determined. Now, to find these coefficients, we substitute the power series back into the differential equation. We also need to differentiate the series term-by-term to get expressions for \( y' \) and \( y'' \). This results in:
\[ y'(x) = \sum_{n=1}^{fty} n a_n x^{n-1} \]
and
\[ y''(x) = \sum_{n=2}^{fty} n(n-1) a_n x^{n-2} \].
Substituting these into the differential equation gives us a new series that we can solve term-by-term for the coefficients \( a_n \).

Elementary Methods

Elementary methods refer to classic techniques for solving differential equations without special functions or series. For \( y'' + y = 4 \sin 3x \), we can solve this using standard approaches. First, we solve the homogeneous part, \( y'' + y = 0 \), which has the characteristic equation \( r^2 + 1 = 0 \). The roots are \( r = \pm i \), giving the general solution \( y_h = C_1 \cos x + C_2 \sin x \). Next, we solve the non-homogeneous part by looking for a particular solution. Due to the right-hand side \( 4 \sin 3x \), we use the method of undetermined coefficients and guess a particular solution of the form \( y_p = A \cos 3x + B \sin 3x \). Upon differentiating and substituting back into the differential equation, we find the values of A and B.

Recurrence Relation

A recurrence relation is an equation that expresses the coefficients in a power series in terms of previous coefficients. From the power series method mentioned earlier, after substituting into the differential equation \( y'' + y = 4 \sin 3x \), we collect like terms so that each coefficient of \( x^n \) must individually equal zero. This provides a recurrence relation that links each coefficient with previous ones:
\[ a_{n+2} = f(a_n, a_{n-1}) \].
Solving this recurrence relation allows us to find a formula for each coefficient \( a_n \). This formula is essential in constructing the complete power series solution for the differential equation.

Method of Undetermined Coefficients

The method of undetermined coefficients is a systematic approach to finding a particular solution to non-homogeneous differential equations. For our example, \( y'' + y = 4 \sin 3x \), we start with the form of the non-homogeneous term on the right side, which is \( \sin 3x \). We then predict a solution called the 'particular solution' of a similar form: \( y_p = A \cos 3x + B \sin 3x \). After this guess, we differentiate \( y_p \) to find \( y_p' \) and \( y_p'' \), and substitute these into the differential equation. Matching coefficients of \( \cos 3x \) and \( \sin 3x \) on both sides allows us to solve for \( A \) and \( B \). This determined particular solution then complements the general solution from the homogeneous equation, giving us the complete solution.