3.5 How to solve a differential equation with initial conditions?

To solve

\[ y''-3y'+2y=10 e^{5 x} \]

with \(y(0)=1,y'(0)=5\) do

eq1:= diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=10*exp(5*x); 
dsolve({eq1,y(0)=1,D(y)(0)=5},y(x)); 
 
Methods for second order ODEs: 
Trying to isolate the derivative d^2y/dx^2... 
Successful isolation of d^2y/dx^2 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 2; linear nonhomogeneous with symmetry [0,1] 
trying a double symmetry of the form [xi=0, eta=F(x)] 
<- double symmetry of the form [xi=0, eta=F(x)] successful 
....
 

The above can also be written using D@@ notation, like this

eq:= (D@@2)(y)(x) - 3*D(y)(x) +2*y(x) = 10*exp(5*x); 
IC := y(0)=1,D(y)(0)=5; 
dsolve({eq,IC},y(x));