Here are some computations that may be useful. Here below, $\big[ \,f\,\big]_a^b$ denotes $f(b)-f(a)$, and $n^{(k)}:=n(n-1)\dots(n-k+1)$.
1. (An integration by parts) Let $w$ be a solution of the third order linear ODE on the interval $[a,b]$: $$(x^4-34x^3+x^2)w''' + 3(2x^3-51x^2+x)w''+(7x^2-112x+1)w'+(x-5)w=0,$$ and put $$M(n):=\int_a^b x^nw(x)dx$$ for any integer $n\ge0$. Then, for any $n\ge 2$, $$n^3M(n)- (34n^3-51n^2+27n-5)M(n-1) + (n-1)^3M(n-2)= \Big[A(x)(x^n)'' + B(x)(x^n)' + C(x)x^n \Big]_a^b,$$ where $$ A:=pw \qquad B:= qw- (pw)' \qquad C:= (pw )'' - (qw)'+ rw ,$$ and $$p(x):=x^3-34x^2+x\qquad q(x):=3x^2-51x\qquad r(x):=x+10-x^{-1}.$$
Proof. We express the above polynomials of $n$ in the base $n^{(k)}$, then we absorb the latter terms as coefficients of derivatives of $x^n$, and finally we integrate by parts.
The RHS writes:$$\int_a^b\Big\{n^3x^nw -(34n^3-51n^2+27n-5)x^{n-1}w +(n-1)^3x^{n-2}w\Big\}dx=$$ $$\int_a^b\Big\{(n^ {(3)}-3n^ {(2)}+2n)x^nw -(34n^{(3)}+51n ^{(2)}+10n-5)x^{n-1}w +(n^{(3)}-n+1)x^{n-2}w\Big\}dx=$$ $$\int_a^b\Big\{(x^n)'''pw+(x^n)''qw+(x^n)'rw+x^n(5x^{-1} -x^{-2} )w\}dx=$$ $$ \int_a^bx^n \Big\{-(pw)'''+(qw)''-(rw)'+ (5x^{-1} -x^{-2} )w\}dx +\Big[ (x^n)''pw- (x^n)'(pw)'+ x^n(pw)'' +(x^n)'qw-x^n(qw)' +rw \Big]_a^b=$$ $$ -\int_a^bx^{n-1} \Big\{(x^4-34x^3+x^2)w''' + 3(2x^3-51x^2+x)w''+(7x^2-112x+1)w'+(x-5)w\Big\}dx+$$ $$+\Big[ (x^n)''pw- (x^n)'(pw)'+ x^n(pw)'' +(x^n)'qw-x^n(qw)' +rw \Big]_a^b=$$ $$=\Big[A(x)(x^n)'' + B(x)(x^n)' + C(x)x^n \Big]_a^b.\qquad\square$$
2.(consequence) Let $w$ a solution of the above linear equation on $(0,c)\setminus\{c_0\}$, with $\int_0^c wdx=1$, and and assume it verifies the following linear boundary conditions, expressed in terms of the above coefficients $a,b,c$:
i) $A(x)=o(1)$, $B(x)=o(1)$ $C(x)=O(1)$, as $x\to0$;
ii) $A(x)$, $B(x)$, $C(x)$, are continuous at $x=c_0$ ,
iii) $A(x)=o(1)$, $B(x)=o(1)$ $C(x)=o(1)$, as $x\to c$.
Then the corresponding $M(n)$ are the Apéry sequence.
Indeed, computing the integral on $[0,c]$ for $M(n)$ as limit of integrals on $[\epsilon, c_0-\epsilon]\cup[c_0+\epsilon, c-\epsilon]$ as $\epsilon\to0$, and applying the above integration by parts formula, one gets that $M(n)$ satisfy the Apéry recurrence, with $M(0)=1$ (note that $M(1)=5M(0)$ follows from the recurrence as well).
3. (Positive solutions of the third order ODE). Assume $u$ solves the second order linear ODE $$(x^3-34x^2+x)u''+(2x^2-51x+1)u'+\frac{1}{4}(x-10)u=0.$$ Then $w:=u^2$ solves $$(x^4-34x^3+x^2)w''' + 3(2x^3-51x^2+x)w''+(7x^2-112x+1)w'+(x-5)w=0.$$
Proof. Put $$P:=x^4-34x^3+x^2\qquad Q:=\frac{x^2}{2}-5x,$$ so the equation for $u$ (multiplied by $2x$) writes: $$2Pu''+P'u'+Qu=0,$$ and we have $$0=(2Pu''+P'u'+Qu)'u+ 3(2Pu''+P'u'+Qu)u'=$$ $$=(2Pu'''+2P'u''+P'u''+P''u'+Qu'+Q'u)u+3(2Pu''+P'u'+Qu)u'=$$ $$=P(2u'''u+6u''u')+3P'(u''u+u'^2)+(P''+4Q)u'u+Q'u^2=$$ $$=Pw''' +\frac{3}{2}P'w''+\Big(\frac{P''}{2}+2Q\Big)w'+Q'w,$$ which is the above third order equation for $w$.