Recently,I'm trying to solve this differential equation: $$ m\frac{d^2x(t)}{dt^2}=(V_1-V_2)\delta(x(t)) $$
The equation has a physical significance:a particle moves from one half-space with potential energy $V_1$ to the other half-space with potential energy $V_2$. So it can be solved by conversation of energy as follows. $$ m\frac{d^2x(t)}{dt^2}\frac{dx}{dt}=(V_1-V_2)\delta(x)\frac{dx}{dt} $$ $$ \frac{1}{2}m\frac{d}{dt}\left(\frac{dx}{dt}\right)^2 = (V_1-V_2)\delta(x)\frac{dx}{dt} $$ Then we can multipy $dt$ to both sides and integral,the equation will be solved easily. So the original equation does make sense.
However,it becomes different if we do not use the conversation of energy. First,we know $$ x(t) = \left\{ \begin{aligned} & at+b & t < -\frac{b}{a} \\ & (k + 1)at+(k + 1)b & t > -\frac{b}{a} \end{aligned} \right. $$ Then, $$ \frac{d^2x}{dt^2}=ka\delta(t + \frac{b}{a}) $$ But it seems that the relation between $\delta(x)$ and $\delta(t + \frac{b}{a})$ is not defined because of the difference of the left and right derivative of $x(t)$ at $t=-\frac{b}{a}$. So the original equation does not make sense.
Why are the two results different? What is the mistake? Thanks for your help.