How to Solve a Second-Order Differential Equation?

I need to solve this equation in matlab:

I wrote this but it gives me an answer with parameter t, which i don't know what means:

eqn2 = 'x*D2y+2*Dy-x*y=2*exp(x)';

pretty(simplify(dsolve(eqn2)))

Answear: C6*exp((t*((x^2 + 1)^(1/2) - 1))/x) - (2*exp(x))/x + C7*exp(-(t*((x^2 + 1)^(1/2) + 1))/x)

I know that the correct answear is

Can you help me with matlab code?

Answer 1

[Paul]

>> syms x y(x)
eqn2 = x*diff(y,x,2) + 2*diff(y,x) - x*y == 2*exp(x);
sol = dsolve(eqn2);
sol = rewrite(sol,'exp');
sol = simplify(sol);
sol = expand(sol);
c = children(sol);
c1 = c(1);
c2 = sum(c(2:end));
[num,den]=numden(c2);
num = expand(num);
num = collect(num,exp(-x));
num = collect(num,exp(x));
num = num/coeffs(den,x);
den = den/coeffs(den,x);
num = collect(collect(num,exp(-x)),exp(x));
sol = c1 + num/den;
pretty(sol)
         sqrt(2) C7 - sqrt(pi)          sqrt(2) C7 + pi sqrt(2) C8
         --------------------- exp(x) + -------------------------- exp(-x)
               2 sqrt(pi)                       2 sqrt(pi)
exp(x) + -----------------------------------------------------------------
                                         x

Make the appropriate substitutions of C1-1/2 and C2 for the coefficients of exp(x) and exp(-x) in the second term on the right. Maybe someone else can get there quicker?

Answer 2

[Alan Stevens]

syms x y
eqn2 = 'x*D2y+2*Dy-x*y=2*exp(x)';
dsolve(eqn2,x)

Warning: Support of character vectors and strings will be removed in a future release. Use sym objects to define differential equations instead.

ans = 

[Filip Bielecki]

But it's not the answer that i wrote in my question, it should be: