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Possibly a Laplace transform approach: $\frac{d^2u}{dx^2}+g(x)\frac{du}{dx}+\frac{du}{dx}=0$
The general form of the equation:
$$
u”+g(x)u’+\frac{du}{dx}=0
$$
where $g(x)$ is some function, does an equation of this form always have two particular solutions that are a linear combination of two linearly independent solutions $cos(kx)$ and $sin(kx)$? More generally,
Is there a closed-form approach to determining how many solutions of
the form of $cos(kx)$ and $sin(kx)$ satisfy the above equation?
Note:
I know how to solve the equation with a symbolic method, and
I know how to find the general solution with a differential/integration method.
A:
$$
\frac{d^2u}{dx^2} + g(x)\frac{du}{dx} + k^2u = 0
$$
Or
$$
\frac{d^2u}{dx^2} + k^2u = 0
$$
Or
$$
\frac{d^2u}{dx^2} + k^2\left(\frac{d^2u}{dx^2} + u \right) = 0
$$
Or
$$
\frac{d^2u}{dx^2} = -k^2u
$$
or
$$
\frac{d^2u}{dx^2} + u = 0
$$
or
$$
\frac{d^2u}{dx^2} + (u – c) = 0
$$
which has solution $u(x) = c_1\cos(kx) + c_2\sin(kx)$.
Authors:
Lidija Preis
The idea for this thesis came in the spring of 2010. While I was doing my master’s degree at the Department of Informatics at the Faculty of Electrical Engineering, KTH Royal Institute
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