Explain what is meant by an integrating factor for an equation of the form

$$\frac{d y}{d x}+f(x, y)=0$$

Show that $2 y e^{x}$ is an integrating factor for

$$\frac{d y}{d x}+\frac{2 x+x^{2}+y^{2}}{2 y}=0$$

and find the solution $y=y(x)$ such that $y(0)=a$, for given $a>0$.

Show that $2 x+x^{2} \geqslant-1$ for all $x$ and hence that

$$\frac{d y}{d x} \leqslant \frac{1-y^{2}}{2 y}$$

For a solution with $a \geqslant 1$, show graphically, by considering the sign of $d y / d x$ first for $x=0$ and then for $x<0$, that $d y / d x<0$ for all $x \leqslant 0$.

Sketch the solution for the case $a=1$, and show that property that $d y / d x \rightarrow-\infty$ both as $x \rightarrow-\infty$ and as $x \rightarrow b$ from below, where $b \approx 0.7035$ is the positive number that satisfies $b^{2}=e^{-b}$.

[Do not consider the range $x \geqslant b$.]