Solve differential equation y'+3y= 3x^2 e^(-3x)

Solve the first-order differential equations:
${\displaystyle (x^2+1)\frac{dy}{dx}=xy}$

Solve differential equation $dy/dx=(y^2-1)/(x^2-1)$

Solve the differential equation $(dP)/dt=4P+a$

I would need some help in solving the following differential equation:
$u^{\prime }(p)+\frac{r+N\lambda p}{N\lambda p(1-p)}u(p)=\frac{r+N\lambda p}{N\lambda (1-p)}g.$
I already know that the solution is of the form
$gp+C(1-p){\left(\frac{1-p}{p}\right)}^{r/\lambda N},$,
where C is a constant.
Now this is a standard linear first-order differential equation. Letting
$f(p)=\frac{r+N\lambda p}{N\lambda p(1-p)},q(p)=\frac{r+N\lambda p}{N\lambda (1-p)}g,\mu (p)=e^{\int f(p)dp},$
the solution should have the form
$u(p)=\frac{C}{\mu (p)}+\frac{1}{\mu (p)}\int \mu (p)q(p)dp,$
where C is a constant.
The calculation of $\mu (p)$ was not much of a problem. Indeed, we can rewrite f(p) as
$f(p)=\frac{r}{N\lambda }\frac{1}{p}+(1+\frac{r}{N\lambda })\frac{1}{1-p}$
Using $\int \frac{1}{p}dp=\ln (p)$ and $\int \frac{1}{1-p}dp=-\ln (1-p)$, I found that
$\mu (p)=p^{\frac{r}{N\lambda }}(1-p{)}^{-1-\frac{r}{N\lambda }}.$
Where I have difficulties is calculating the term ∫μ(p)q(p)dp, where
$\mu (p)q(p)=\frac{rg}{N\lambda }{p}^{\frac{r}{N\lambda }}(1-p{)}^{-2-\frac{r}{N\lambda }}+p^{\frac{r}{N\lambda }+1}(1-p{)}^{-2-\frac{r}{N\lambda }}.$
How do I integrate this last term? Or did I miss something obvious, because in the paper I'm looking at they don't provide any details about how they get to the solution of the differential equation
Edit: small typo corrected in the last displayed equation.
Edit: there is a big typo in the differential equation: the rhs should have $r+N\lambda$ rather than $r+N\lambda p$ in the numerator. This probably makes it much simpler to solve. My bad.

Solve differential equation$x{y}^{\prime }=(1-y^2{)}^{\frac{1}{2}}$

I am curious if there are any examples of functions that are solutions to second order differential equations, that also parametrize an algebraic curve.
I am aware that the Weierstrass $\mathrm{\wp }$ - Elliptic function satisfies a differential equation. We can then interpret this Differential Equation as an algebraic equation, with solutions found on elliptic curves. However this differential equations is of the first order.
So, are there periodic function(s) $F(x)$ that satisfy a second order differential equation, such that we can say these parametrize an algebraic curve?
Could a Bessel function be one such solution?

Given the function $\{\begin{array}{ll}{e}^{-t}& \text{if }0\le t<2\\ 0& \text{if }2\le t\end{array}$
Express f(t) in terms of the shifted unit step function u(t -a)
F(t) - ?
Now find the Laplace transform F(s) of f(t)
F(s) - ?