Examples of second order linear differential equation with non-trivial solutions

Please give me some non-trivial examples of differential equations with second order that is linear.

A:

A differential equation $y” + p(x)y’ + q(x)y = 0$ is linear if and only if both $p$ and $q$ are linear. The nonlinear case is
$$y”+p(x)y’ + q(x)y=0\tag{1}$$
$$y”+p(x)y’+q(x)y’ =0\tag{2}$$
In particular, equations $(1)$ and $(2)$ are both linear. From Sturm-Liouville theory, we know that there exists a non-trivial solution to $(1)$ if and only if
$$\int_0^{+\infty}x^m q(x)dx = 0,\qquad \int_0^{+\infty}x^mp(x)dx \gt 0$$
and a non-trivial solution to $(2)$ if and only if
$$\int_0^{+\infty}x^mq(x)dx \gt 0,\qquad \int_0^{+\infty}x^mp(x)dx = 0$$
For example, the particular case where $p(x) = x$ and $q(x)$ is an odd function is
$$y”+xy=0$$
but this differential equation does not have a non-trivial solution because there’s an infinite limit at $+\infty$ and the corresponding integral is $0$
The particular case where $p(x) = ax$, $q(x) = 1$ and $a$ is a positive real number is
$$y” + ay = 0$$
and this equation does not have a non-trivial solution, because the integral $\int_0^{+\infty}x^ma^xdx$ is $0$ and there’s not enough “good” kind of derivatives.
Now, for equation $(2)$, we can start with a non-trivial solution of $(1)$ and apply the differential operator \$A(x

0.0000002
Suppose -p + 2*p = 7. Suppose p = i – 4. Suppose i = 2*a – 37. What is a rounded to the nearest 10?
30
Let w be (-4)/12*(1 – -1). Let i be w/1 – 3/(-3). Let h be (-20*i)/(4/9900). Round h to the nearest ten thousand.
-110000
Let d(v) = v**3 + 3*v**2 – v + 2. Let f be d(-3). Let i = f + -2. Let j = 6 + i. Round j to the nearest 10.
10
Let c = 7.6 + -11.71. Let j = 7 – 13. Let v = c – j. Round v to 1 dp.
-0.1
Suppose -4*r + 24 = -0*r. Let n be r/(-4)*(0 + -60). Round n to the nearest 10.
100
Let w = 5.95 + -5. Let u = -0.8 + 0.7. Let j = u – w. What is j rounded to the nearest integer?
-1
Let j = -30 –
d0c515b9f4

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SOLR 6 indexing legacy data – seq id

I am trying to combine multiple SOLR instances to form a single index. Each instance stores the same kind of data and the name of the different collections and SOLR instances do not vary much. The general rule is that we add multi-collections (eg collections such as accounts, orders, customers) in a list and each collection corresponds to a separate instance of SOLR. The index “index” is the same across multiple SOLR instances. The structure of the collection document looks like:
{
“collections”: [
{
“collection_name”: “collection1”,
“docs” : [
{
“id”: 1,
“name”: “doc_name1”,

FcteA

12-08 20:08

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