Toronto Math Forum

Solve $y'' - y' - 2y = cosh2t$

Firstly, consider homogeneous equation $y'' - y' - 2y = 0$, the characteristic polynomial is
\begin{align*}
r^2 - r - 2 &= 0\\
(r + 1)(r - 2) &= 0
\end{align*}

Therefore, the solution to homogeneous equation is $c_1 e^{-t} + c_2 e^{2t}$

Then need to solve $y'' - y' - 2y = cosh2t$, since
\begin{equation*}
    cosh(2t) = \frac{e^{2t} + e^{-2t}}{2}
\end{equation*}

we come to guess $y = Ate^{2t} + Be^{-2t}$ is a solution.
\begin{align*}
    y' &= Ae^{2t} + 2Ate^2t - 2Be^{-2t}\\
    y'' &= 2Ae^{2t} + 2Ae^{2t} + 4Ate^{2t} + 4Be^{-2t}\\
    &= (4A + 4At)e^{2t} + 4Be^{-2t}
\end{align*}
then plug $y$, $y'$ and $y''$ back in $y'' - y' - 2y$, we get

\begin{align*}
    y'' - y' - 2y =& (4A + 4At)e^{2t} + 4Be^{-2t} - (Ae^{2t} + 2Ate^2t - 2Be^{-2t}) - 2(Ate^{2t} + Be^{-2t}) \\
    &= (4A - A)e^{2t} + (4A - 2A - 2A)te^{2t} + (4B + 2B - 2B)e^{-2t}\\\
    &= 3Ae^{2t} + 4Be^{-2t}
\end{align*}
since $y'' - y' - 2y = cosh2t$
\begin{align*}
    &3Ae^{2t} + 4Be^{-2t} = \frac{e^{2t} + e^{-2t}}{2}\\
    &\implies 3A = \frac{1}{2}\quad and\quad 4B = \frac{1}{2}\\
    &\implies A = \frac{1}{6}\quad and \quad B = \frac{1}{8}
\end{align*}
Therefore the solution to $y'' - y' - 2y = cosh2t$ is $y = \frac{1}{6}te^{2t} + \frac{1}{8}e^{-2t}$
Then, the general solution is
\begin{equation*}
    y = c_1 e^{-t} + c_2 e^{2t} + \frac{1}{6}te^{2t} + \frac{1}{8}e^{-2t}
\end{equation*}

Solve \begin{align*}
    \frac{x dx}{(x^2 + y^2)^\frac{3}{2}} + \frac{y dy}{(x^2 + y^2)^\frac{3}{2}} = 0
\end{align*}

We can multiple both side by $(x^2 + y^2)^\frac{3}{2}$, then we have
\begin{equation}
    xdx + ydy = 0
\end{equation}

Let $M = x$ and $N = y$
\begin{equation}
    M_y = 0\quad \quad N_x = 0
\end{equation}

Since $M_y = N_x$,  the equation is exact so $\exists \psi(x, y)\quad s.t \quad \frac{\partial \psi}{\partial x} = M$ and $\frac{\partial \psi}{\partial y} = N$
Therefore
\begin{align*}
    \psi(x, y) &= \int M dx \\
    &= \int x dx\\
    &= \frac{1}{2}x^2 + h(y)\\
\end{align*}
Since
\begin{align*}
    \frac{\partial \psi}{\psi y} &= h'(y)\\
    \implies h'(y) &= y\\
    \implies h(y) &= \frac{1}{2}y^2 + C
\end{align*}
Therefore, the solution is $\frac{1}{2}x^2 + \frac{1}{2}y^2 = C$, rearrange we get $x^2 + y^2 = C$, where $x \neq 0$ and $y \neq 0$