.1  and then use  h = 0.05. y' = 3x − 4y + 3, y(1) =

Here, in the question given the set of equations as: $y\text{'} = 3x-4y+3$. We have to calculate the four-decimal approximation of the indicated value. Now first we shall know about the improved Euler's method. In Euler's method, we approximate the function by a rectangular shape ( shown below):

To improve the approximation we use the improved approximation method. 

Steps in Improved Euler's method:

Step 1:

 $K_1 = f( t_n, y_n)$

Step 2:

$K_2=f(t_n+h, y_n+h{k}_1)$

Step 3:

$y_{n+1}= y_n+h\frac{k_1+k_2}{2}$