Solving simple differential equations

1. Solve an ordinary differential equation $\frac{dy}{dt}=ay$

   syms y(t) a;
   eqn = diff(y, t) == a*y;
   S = dsolve(eqn)
   

   Add conditions , initial value $y(0)=5$

   syms y(t) a;
   eqn = diff(y, t) == a*y;
   cond = y(0) = 5;
   S = dsolve(eqn, cond)
   

2. Solving second order ordinary differential equations $\frac{d^{2}y}{d{u}^2}=ay$

   syms y(t) a;
   eqn = diff(y, t, 2) == a*y;
   S = dsolve(eqn)
   

   Add conditions , $y(0)=b,y^{\prime }(0)=1$

   syms y(t) a b;
   eqn = diff(y, t, 2) == a*y;
   Dy = diff(y, t);
   cond = [y(0) == b, Dy(0) == 1];
   S = dsolve(eqn, cond)
   

3. Solving ordinary differential equations
   $$\begin{gathered} \frac{dy}{dt}=z \\ \frac{dz}{dt}=-y \end{gathered}$$

   syms y(t) z(t)
   eqns = [diff(y, t) == z, diff(z, t) == -y];
   s = dsolve(eqns)