Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Discrete mathematics, continuous mathematics and probability / 4

Let $I=\int e^x\sin xdx$. Two applications of integration by parts givewhere additive constants can be absorbed at the end. Solving for $I$ yieldsDifferentiating the answer gives $e^x\sin x$.

After division by $x>0$, the equation is linear with coefficient $-1/x$ on $y$. Its integrating factor is $1/x$, and the product rule givesUsing the antiderivative from the preceding part,The arbitrary constant multiplies the homogeneous solution $x$.

The homogeneous characteristic equation is $r^2-4r+3=(r-1)(r-3)=0$, giving $e^x$ and $e^{3x}$. Since the forcing $e^{3x}$ already appears among these solutions, try a particular solution $y_p=Ax{e}^{3x}$. Differentiation and substitution giveso $A=1/2$. The general solution is

Write $Y(t)=y(e^t)$, so $t=\log x$. The chain rule givesSubstitution into the Cauchy-Euler equation therefore gives $Y^{\prime \prime }+Y^{\prime }-2Y=0$. Its characteristic equation factors as $(r-1)(r+2)=0$, so $Y=C_1{e}^t+C_2{e}^{-2t}$. Returning to $x$ gives