Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 2 / 5C / a / Solution

The Wronskian is $W=y_1{y}_2^{\prime }-y_2{y}_1^{\prime }$. Abel's identity givesIf $W\ne 0$, the two column vectors in the displayed representation form a basis of ${\mathbb{R}}^2$ at each $x$. Differentiating the first component and comparing it with the second givesSubstitution in the differential equation givesSolving this two-by-two system yields