 Solution

The characteristic polynomial of the homogeneous equation factors as $(r-1)(r-3)$, giving the complementary function $C_1{e}^x+C_2{e}^{3x}$. For a particular solution take $A\cos x+B\sin x$. Comparing cosine and sine coefficients gives $2A-4B=10$ and $4A+2B=0$, hence $A=1,B=-2$. ThereforeThe initial conditions imply $C_1+C_2=2$ and $C_1+3C_2=2$, so $C_1=2,C_2=0$. The required solution isIt has $y(0)=3$ and $y^{\prime }(0)=0$, and direct differentiation verifies $y^{\prime \prime }-4y^{\prime }+3y=10\cos x$.