Question

Determine the general solution of $$ y^{\prime \prime}+\lambda^{2} y=\sum_{m=1}^{N} a_{m} \sin m \pi t $$ $$ \text { where } \lambda>0 \text { and } \lambda \neq m \pi \text { for } m=1, \ldots, N $$

Short answer

Answer: The general solution for the given inhomogeneous equation is $$y(t) = A\cos(\lambda t) + B\sin(\lambda t) + \sum_{m=1}^N \frac{a_m}{\lambda^2 - m^2 \pi^2}\sin m\pi t$$.

Step-by-step solution

Find the complementary solution for the Homogeneous equation

The given differential equation is inhomogeneous, so let's start by finding the complementary solution for the homogeneous equation: $$ y^{''}+\lambda^2 y = 0 $$ This is a second order linear homogeneous differential equation with constant coefficients and has the auxiliary equation: $$ r^2 + \lambda^2 = 0 $$ Solving this equation, we get: $$ r = \pm \lambda i $$ So, the complementary solution (the general solution for the homogeneous) is given by: $$ y_c(t) = A\cos(\lambda t) + B\sin(\lambda t) $$ Where A and B are constants.

Finding a particular solution for the inhomogeneous equation

Now, we need to find a particular solution for the inhomogeneous equation. Since the given inhomogeneous equation is a sum of sines, we can find the solution for each independent term and then sum them up. Consider a single term in the inhomogeneous source: $$ a_m \sin m\pi t $$ We can find a particular solution for this term by suggesting a solution of the form: $$ y_p(t) = C_m\sin m\pi t $$ Plug this solution into our inhomogeneous equation: $$ -y_p'' + \lambda^2 y_p = a_m \sin m\pi t $$ We get: $$ (C_m m^2\pi^2 + \lambda^2 C_m)\sin m\pi t = a_m \sin m\pi t $$ Notice here that $$\lambda^2 \neq m^2\pi^2$$, so we can solve for $$C_m$$ without encountering division by zero: $$ C_m = \frac{a_m}{\lambda^2 - m^2 \pi^2} $$ Thus, $$ y_p(t) = \frac{a_m}{\lambda^2 - m^2 \pi^2}\sin m\pi t $$

Summing up the particular solutions

Now, sum up the particular solutions for all terms: $$ y_p(t) = \sum_{m=1}^N \frac{a_m}{\lambda^2 - m^2 \pi^2}\sin m\pi t $$

Combine complementary and particular solutions

Finally, we'll combine the complementary and particular solutions to find the general solution of the inhomogeneous equation: $$ y(t) = A\cos(\lambda t) + B\sin(\lambda t) + \sum_{m=1}^N \frac{a_m}{\lambda^2 - m^2 \pi^2}\sin m\pi t $$ This is the general solution of the given differential equation.

Key concepts

Complementary Solution

In the context of differential equations, the complementary solution represents the part of the solution that solves the associated homogeneous equation. This homogeneous equation is obtained by setting the function that accounts for any external forces (or inhomogeneous terms) to zero. For instance, given the equation \( y'' + \lambda^2 y = 0 \), we only focus on the left-hand side, deliberately ignoring non-homogeneous or forced components.

To find the complementary solution for \( y'' + \lambda^2 y = 0 \), we employ methods that solve linear homogeneous differential equations with constant coefficients. The auxiliary equation \( r^2 + \lambda^2 = 0 \) guides us in figuring out the characteristic values, which are \( r = \pm \lambda i \). These complex roots imply an oscillatory form in the solution, resulting in a complementary solution of:

• \( y_c(t) = A\cos(\lambda t) + B\sin(\lambda t) \)

Here, \( A \) and \( B \) are arbitrary constants, determined by initial conditions or boundary values if specified.

Homogeneous Equation

A homogeneous equation typically refers to a differential equation without any "extra" functions added to the system. Mathematically, it appears as an equation where every term is a function or derivative of the unknown function. The homogeneous part of the equation \( y'' + \lambda^2 y = 0 \) disregards any external forces or inhomogeneous terms. This leads us to work with natural behaviors produced by the system itself.

Homogeneous equations allow us to understand and solve for inherent characteristics of the system, like oscillations or decay patterns, that come from the properties of the differential operator itself. By concentrating solely on the homogeneous part, we position ourselves to derive solutions using characteristic equations, yielding general solutions comprised of exponentials or trigonometric functions depending on the roots of the characteristic equation.

Particular Solution

The particular solution specifically addresses the inhomogeneous equation components. In our problem, the inhomogeneity comes from \( \sum_{m=1}^{N} a_{m} \sin m \pi t \), implying periodic external forces acting on the system. Deriving the particular solution involves seeking a function that satisfies the complete differential equation as given, including the non-homogeneity.

To find the particular solution for a term like \( a_m \sin m\pi t \), assume a potential solution form such as \( y_p(t) = C_m \sin m\pi t \). Substituting back into the differential equation and matching coefficients helps in determining \( C_m \). We deduce:

• \( C_m = \frac{a_m}{\lambda^2 - m^2 \pi^2} \)

This ensures that the proposed form aligns perfectly with the inhomogeneous part of the equation. The complete particular solution is then:

• \( y_p(t) = \sum_{m=1}^{N} \frac{a_m}{\lambda^2 - m^2 \pi^2} \sin m\pi t \)

This step-by-step approach provides individual terms that we eventually sum up to represent the total external force response.

Inhomogeneous Equation

Inhomogeneous equations are differential equations that include some external force or bias in addition to the regular terms of the equation. The inhomogeneity typically appears as an added function, such as the sum of sine functions \( \sum_{m=1}^{N} a_{m} \sin m \pi t \) found in this differential equation problem.

Dealing with inhomogeneous equations involves accounting not only for the inherent system properties but also for how these external factors disturb or alter the system's behavior. The strategy is to decompose the general solution into the complementary solution, which handles the homogeneous part, and the particular solution, capturing the effects of the added non-homogeneous function. Therefore, a general solution appears as follows:

• \( y(t) = A\cos(\lambda t) + B\sin(\lambda t) + \sum_{m=1}^{N} \frac{a_m}{\lambda^2 - m^2 \pi^2}\sin m\pi t \)

This solution structure effectively integrates both natural system responses and external influences, offering a complete picture of the system dynamics.