STEP 3 2018 Q3 Question

Hello, as the title suggests, I wanted to ask about a particular STEP question, specifically finding equation (*). The mark scheme just wants you to demonstrate solutions exist by solving simultaneous equations, but why is (*) assumed? As it does not seem to make intuitive sense, so I was wondering if anybody could guess why we assume it can be written as nested derivatives?

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mqb2766

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Original post by MathsEnjoyer

Hello, as the title suggests, I wanted to ask about a particular STEP question, specifically finding equation (*). The mark scheme just wants you to demonstrate solutions exist by solving simultaneous equations, but why is (*) assumed? As it does not seem to make intuitive sense, so I was wondering if anybody could guess why we assume it can be written as nested derivatives?

If you had
(x^2 y)''
then that would be a particular case of the lhs of the ode, so something like
x^2 y'' + 4xy' + 2y
Then just integrate twice and divide by x^2. The form they give is similar, although more general.