QUESTION $3$

$(3\mathrm{.}1)$ Solve the following differential equation:

$(6)$

$y^{\text{'}\text{'}\text{'}}-y^{\text{'}\text{'}}=0$

$(3\mathrm{.}2)$ Show that

$(2xy-se{c}^{2}x)dx+(x^2+2y)dy=0$

is an exact differential equation, and hence find a solution in the form $f(x,y)=c.$

$(3\mathrm{.}3)$ Consider the Bernoulli differential equation

$\frac{dy}{dx}-5y=-\frac{5}{2}x{y}^3$

$on$ xin$(-,)$

$($a$)$ Show that an appropriate substitution leads to a first order linear differential equation of the form

$(7)$

$\frac{du}{dx}+10u=5x$

List any solutions of equation $()$ excluded by your answer.