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

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

                            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