Solving a second order differential equation - Force depending on displacement

Question

I know that an object starts moving along the $y$ axis with initial speed $v_o$ and that the force acting on this object is $$F(t) = -c\cdot y(t)$$ I need to find the expression for $y$ in terms of time.
Therefore, $$F(t) = m \frac{dv}{dt} = m \frac{d^2 y}{dt^2}$$ $$m \frac{d^2y}{dt^2} = c \cdot y(t)$$ $$\frac{1}{y(t)} d^2y = \frac{c}{m} dt^2 $$
I have never studied differential equations before, I came across this problem when I during leisurely-reading of a Physics book. Can this equation be solved in an easy, quite intuitive way?

Answer 1

You have two mistakes - should be $F(t)=ma=m\dfrac{\text{d}v}{\text{d}t}$ and $m\dfrac{\text{d}^2y}{\text{d}t^2}=-cy(t)$. The equation is $$my''+cy=0$$ We guess a solution of the form $y=e^{\lambda t}$ and the equation becomes $$m\lambda^2e^{\lambda t}+ce^{\lambda t}=0\implies \lambda^2+\frac{c}{m}=0$$ Solving the quadratic equation you will have two solutions. Each linear combination of them is a solution to the equation.