TITLE: Simple Differential Equations Question.
QUESTION [0 upvotes]: The question reads:
Let $C_1$ and $C_2$ be arbitrary constants. For each of the following $y(x)$, find a differential equation that y satisfies, with the order of the ODE equal to the number of arbitrary constants in $y(x)$. The ODE should not contain any arbitrary constants.
(a)   $y(x) = cos(x) + C_1x^2$
Could someone explain what is really going on in this question because I can't seem to get the intuition behind it. Thanks in advance for your time and help.

REPLY [0 votes]: Isolate the constant, for example with
$$\frac{y(x)}{x^2}=\frac{\cos x}{x^2}+C_1.$$
Then taking the derivative, you eliminate it.
$$\left(\frac{y(x)}{x^2}\right)'=\left(\frac{\cos x}{x^2}\right)'.$$
Now you can expand/reshape this equation (an ODE) the way you like. For example, the linear form
$$x^2y'(x)-2x\,y(x)=-x^2\sin x-2x\cos x.$$

Another example with two constants:
$$y=ax^b.$$
Isolate $a$ by taking the logarithm:
$$\log y=a+b\log x.$$
Differentiate:
$$\frac{y'}y=\frac bx.$$
Isolate $b$:
$$\frac{xy'}y=b.$$
Differentiate:
$$\left(\frac{xy'}y\right)'=0.$$
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