Is there a way to solve this problem using Laplace transforms when only one initial condition is given? I've gotten to a point in the problem where I have
$$-s^2X'(s) +(1-5s)X(s) = x''(0)$$
Is there a way to solve for $x(t)$ from here?
Is there a way to solve this problem using Laplace transforms when only one initial condition is given? I've gotten to a point in the problem where I have
$$-s^2X'(s) +(1-5s)X(s) = x''(0)$$
Is there a way to solve for $x(t)$ from here?
I don't know where you get the $x''(0)$. The Laplace transform of your differential equation should be $$ -s^2 X'(s) + (1-5 s) X(s) + 4 x(0) = 0$$ After plugging in $x(0)=0$, you have a first-order differential equation to solve. Of course the solution will have a free parameter, because your single initial condition is not enough to determine a solution.
This is a linear first order equation of the form $$y' + p(x)y = q(x).$$ Multiply both sides by $\mu(x) = \exp\{\int p(x)\, dx\}$. Undo the product rule on the left hand side and integrate.