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.