General solution We consider the differential equation \[\frac{\dd y}{\dd t} = t\,y\] Using the method of separation of variables, you can find a general solution \[y(t)=c\cdot e^{\frac{1}{2}t^2}\] for some constant \(c\). MATLAB can determine this solution via the dsolve function:

>> syms y(t)  % make a symbolic function y(t)
>>            % use == as symbol for an equation and diff for derivative
>> GDV = diff(y,t) == t*y   
GDV(t) =
diff(y(t), t) == t*y(t)
>> ysol = dsolve(GDV)   % solve ODE exactly
ysol =
C1*exp(t^2/2)

Initial Value Problem In the dsolve command you can also specify the initial value as an equation. The example is the initial value problem \[\frac{\dd y}{\dd t} = y^2,\quad y(0)=1\] Via the method of separation of variables, you can find a general solution \[y(t)=\frac{1}{c-t}\] for some constant \(c\). From \(y(0)=1\) follows that \(c=1\). MATLAB can determine this solution via the dsolve function:

>> syms y(t)
>> GDV = diff(y,t) == y^2
GDV(t) =
diff(y(t), t) == y(t)^2
>> cond = y(0) == 1;
>> ysol = dsolve(GDV, cond)
ysol =
-1/(t - 1)

ODE of order 2 The example is the initial value problem \[\frac{\dd^2y}{\dd t^2} -3\frac{\dd y}{\dd t}+2y=0,\quad y(0)=-1,\quad \frac{dy}{dt}(0) = -3\] In the next section we will learn how to solve this problem, but the solution is \[y(t)=e^t - 2e^{2t}\]

>> syms y(t) 
>> Dy = diff(y,t) 
Dy(t) = 
diff(y(t), t) >> GDV = diff(y,t,2) - 3*diff(y,t) + 2*y == 0 
GDV(t) = 
2*y(t) - 3*diff(y(t), t) + diff(y(t), t, t) == 0 
>> cond = [y(0)==-1, Dy(0)==-3]; 
>> ysol = dsolve(GDV, cond) 
ysol = 
exp(t) - 2*exp(2*t)

We continue this example to show how you can compute further with the symbolic solution in numerical sense. Substituting values of \(t\) does not work straightforwardly because yopl is a symbolic expression or rather a string. Use eval to actually execute the string command. If you want to quickly calculate in MATLAB and make use of vectorisation use, then you must indicate this explicitly.

>> t = [0,1,2]
t =
     0     1     2
>> ysol(t)
Subscript indices must either be real positive integers or logicals.
Error in sym/subsref (line 841)
            R_tilde = builtin('subsref',L_tilde,Idx); 
>> ysol  % this is actually a string object
yopl =
exp(t) - 2*exp(2*t)
>> eval(yopl)    % carry out the string command
ans =
   -1.0000  -12.0598 -101.8072
>> eval(vectorize(ysol))  % use vectorisation
ans =
   -1.0000  -12.0598 -101.8072