Q.Find the differential equation of the family of curves $y=\dfrac{a}{x}+b$ where $a$ and $b$ are arbitrary constants.

Differentiating $bx+ay=ab$ once gives a constant slope, and differentiating again gives $\frac{d^2y}{dx^2}=0$ — so the statement is true.

The given family is

$bx + ay = ab,$

with $a$ and $b$ as arbitrary constants. Differentiate both sides with respect to $x$:

$b + a\frac{dy}{dx} = 0 \quad\Rightarrow\quad \frac{dy}{dx} = -\frac{b}{a},$

which is a constant. Differentiating once more with respect to $x$: …

Differentiating $y = k_1 e^x + k_2 e^{-x}$ twice gives $\dfrac{d^2y}{dx^2} = k_1 e^x + k_2 e^{-x} = y$, i.e. $\dfrac{d^2y}{dx^2} - y = 0$.

Start from

$y = k_1 e^x + k_2 e^{-x}.$

First derivative:

$\frac{dy}{dx} = k_1 e^x - k_2 e^{-x}.$

Second derivative:

$\frac{d^2y}{dx^2} = k_1 e^x + k_2 e^{-x}.$

The right-hand side is exactly $y$ again, so …

A solution obtained from the general solution by giving definite values to the arbitrary constants is called a particular solution.

The general solution of a differential equation contains as many arbitrary constants as the order of the equation. When these arbitrary constants are assigned particular (specific) values — usually determined from given initial or boundary conditions — …

The form $e^{-2x}(A\cos x+B\sin x)$ means complex roots $-2\pm i$; forming the auxiliary equation gives $y_2+4y_1+5y=0$.

The general solution $y=e^{-2x}(A\cos x+B\sin x)$ corresponds to a linear differential equation with complex conjugate roots of the form $\alpha\pm i\beta$, where $\alpha=-2$ and $\beta=1$. So the roots are:

$m=-2\pm i.$

The auxiliary (characteristic) equation is $(m-(-2+i))(m-(-2-i))=0$, i.e.

$(m+2)^2-(i)^2=0\;\Rightarrow\;(m+2)^2+1=0.$ …

Differentiate $y^2 = (x+c)^3$ to get $2y\dfrac{dy}{dx} = 3(x+c)^2$, then eliminate $c$ using the original relation to obtain $8\left(\dfrac{dy}{dx}\right)^3 = 27y$.

The general solution $y^2 = (x+c)^3$ contains one arbitrary constant $c$, so the differential equation is obtained by differentiating once and eliminating $c$.

Differentiate both sides with respect to $x$:

$2y\frac{dy}{dx} = 3(x+c)^2.$

From this, $(x+c)^2 = \dfrac{2y}{3}\dfrac{dy}{dx}$. Cube both sides:

$(x+c)^6 = \left(\frac{2y}{3}\frac{dy}{dx}\right)^3 = \frac{8y^3}{27}\left(\frac{dy}{dx}\right)^3.$

…