How to modify an LTI differential equation to be acausal or anti-causal?
It's important to realize that generally the differential equation (DE) alone doesn't tell us anything about causality. You claim that the system given in your question is causal. However, an anti-causal system with impulse response $h(t)=-u(-t)$ (where $u(t)$ is the step function) also satisfies the very same DE! So the given DE actually describes two systems, a causal one and an anti-causal one.

What you need to enforce causality (and linearity and time-invariance) is the auxiliary condition that the system is initially at rest. "Initially" means the moment in time when the input signal becomes non-zero. If we denote this moment by $t_0$, these conditions are

$$y(t_0)=y'(t_0)=y''(t_0)=\ldots =0\tag{1}$$

where $y'(t)$ denotes the first derivative of the output signal, etc. Solving the DE with the initial rest conditions $(1)$ for $t&gt;t_0$ gives you the output signal of the causal system described by the DE.

A similar thing is true when using the Laplace transform to compute the transfer function of an LTI system. The expression for the Laplace transform of the transfer function does not uniquely identify the corresponding system. You also have to specify the region of convergence (ROC). In your example, the system's transfer function is

$$H(s)=\frac{1}{s}\tag{2}$$

Depending on the ROC, this function corresponds to two different time domain functions:

$$h(t)=u(t),\quad \text{Re}(s)&gt;0$$
and
$$h(t)=-u(-t),\quad \text{Re}(s)&lt;0$$

You need to choose which solution makes sense in your case.