Krishna
0

Step 1: Take the given differential equation, perform the integration on both sides and simplify it for the expression in variables (x and y)

EXAMPLE: \frac{dy}{dx} = 3x - \frac{5}{\sqrt{x}} - 2

Apply the integration on both sides

$\int \frac{dy}{dx} = \int [3x - \frac{5}{\sqrt{x}} - 2]$

y\ =\frac{3\ x^{1+1}}{1+1}\ -\ \frac{5\ x^{\frac{-1}{2}+1}}{\frac{-1}{2}+1}\ -2x\ +\ c

y\ =\ \frac{3\ x^2}{2}\ -\ \frac{5\ x^{\frac{1}{2}}}{\frac{1}{2}}\ -\ 2x\ +c

Step 2: Substitute the given point in the expression to form a linear equation for c.

EXAMPLE: Given point (4, 5) must use x = 4 and y = 5 substitute in the

expression

5\ =\ \frac{3}{2}\left(16\right)\ -\ 10\ \cdot2\ -8\ +\ c

Step 3: Simplify the linear equation for the "c" value

NOTE: Apply the BODMAS rules

Step 4: Find the f(x) by substituting the "c" value in the expression got in step 1

EXAMPLE: f(x) = y or f(4) = 5

f(x) =  y\ =\ \frac{3\ x^2}{2}\ -\ \frac{5\ x^{\frac{1}{2}}}{\frac{1}{2}}\ -\ 2x\ +c

f(x) =\ \frac{3\ x^2}{2}\ -\ 10\ x^{\frac{1}{2}}\ -\ 2x\ +\ 9 (since c = 9)

(ii) Find an equation of the tangent to C at the point P,

Step 1: Calculate the slope(gradient) of the equation

[NOTE: differential coefficient (dy/dx) - what we call gradient.]

EXAMPLE:  f(x) =\ \frac{3\ x^2}{2}\ -\ 10\ x^{\frac{1}{2}}\ -\ 2x\ +\ 9

Apply the differentiation on both sides

\frac{dy}{dx} = m = 3*4 - \frac{5}{2} - 2

m = 15/2 = 7.5

Step 2:Find the equation of a line given one point and the slope

FORMULA : y - y_1 = n(x - x_1)

Substitute the values in the formula and simplify for the equation

EXAMPLE: y - 5 = \frac{15}{2}(x - 4)

Step 3: Simplify the equation