I want to treat w as a constant

syms y(x)
ode = diff(y,x,2) +2*diff(y,x) +(w^2+1)*y == 0;
Dy = diff(y);
cond1 = y(0) == 1
cond2 = Dy(0) == 0
conds = [cond1 cond2]
ySol(x) = dsolve(ode,conds);
ySol = simplify(ySol)

the answer is

exp(-x)*(cos(wx)+1/w*sin(wx))

how can I get the answer?

Answer 1

[David Goodmanson]

Hello HJ,

try replacing the first line with

syms y(x) w real

in which case the result is

ySol(x) =
(exp(-x)*(sin(w*x) + w*cos(w*x)))/w

Answer 2

[Walter Roberson]

syms y(x)
syms w real
ode = diff(y,x,2) +2*diff(y,x) +(w^2+1)*y == 0;
Dy = diff(y);
cond1 = y(0) == 1;
cond2 = Dy(0) == 0;
conds = [cond1 cond2]

conds = 

ySol(x) = dsolve(ode,conds);
ySol = simplify(ySol)

ySol(x) = 

To get from that to what you want is sightly tricky. MATLAB has "preferred" ways to express terms, and you need to force it to overwride that preferred forms. That is, the output is already equivalent to what you want, but to force it to be written the same way you want you need to take a couple of steps that are not obvious.

ch = children(ySol(x));
ch{2}*expand(ch{1}*ch{3})

ans =