Problem 05
$2y \, dx = 3x \, dy$,   when   $x = -2$,   $y = 1$.
 

Solution 05
From Solution 04,
$\dfrac{x^2}{y^3} = c$
 
 

Problem 03
$xy \, y' = 1 + y^2$,   when   $x = 2$,   $y = 3$.
 

Solution 03
$xy \, y' = 1 + y^2$

$xy \dfrac{dy}{dx} = 1 + y^2$
 

Problem 02
$2xy \, y' = 1 + y^2$,   when   $x = 2$,   $y = 3$.
 

Solution 2
$2xy \, y' = 1 + y^2$

$2xy \dfrac{dy}{dx} = 1 + y^2$

Problem 04
$2y \, dx = 3x \, dy$,   when   $x = 2$,   $y = 1$.
 

Solution 04
$2y \, dx = 3x \, dy$

$\dfrac{2y \, dx}{xy} = \dfrac{3x \, dy}{xy}$
 

Problem 340
For the system of pulleys shown in Fig. P-340, determine the ratio of W to P to maintain equilibrium. Neglect axle friction and the weights of the pulleys.
 

Problem 339
The differential chain hoist shown in Fig. P-339 consists of two concentric pulleys rigidly fastened together. The pulleys form two sprockets for an endless chain looped over them in two loops. In one loop is mounted a movable pulley supporting a load W. Neglecting friction, determine the maximum load W that can just be raised by a pull P supplied as shown.
 

Quadratic Equation
Quadratic equation is in the form
 

Where
a, b, & c = real-number constants
a & b = numerical coefficient or simply coefficients
a = coefficient of x²
b = coefficient of x
c = constant term or simply constant
a cannot be equal to zero while either b or c can be zero
 

The Expansion of (a + b)ⁿ
If   $n$   is any positive integer, then

$(a + b)^n = a^n + {_nC_1}a^{n - 1}b + {_nC_2}a^{n - 2}b^2 + \, \cdots \, + {_nC_m}a^{n - m}b^m + \, \cdots \, + b^n$
 

Where
${_nC_m}$ = combination of n objects taken m at a time.
 

Special Products
1.   $(x + y)(x - y) = x^2 - y^2$

2.   $(x + y)^2 = x^2 + 2xy + y^2$

3.   $(x - y)^2 = x^2 - 2xy + y^2$