$$Let \space f(x,y)= \dfrac {dy}{dx}=\dfrac {y^2-x^2}{y^2+x^2}$$

Part I

Here $h=0.2 , x_0 = 0$ & $y_0 = 1$

By Runge kutta method of 4th order

$$k_1=hf(x_0,y_0)=0.2\times f(0,1)=0.2(\dfrac {1^2-0}{1^2+0})=0.2$$ $ k_2=hf(x_0+\dfrac h2,y_0+\dfrac {k_1}2)\\ = hf(0+\dfrac {0.2}2,1+\dfrac {0.2}2)\\ =0.2\dfrac {(1.1^2-0.1^2)}{(1.1^2+0.1^2)}=0.1967 \\ k_3=hf(x_0+\dfrac h2,y_0+\dfrac {k_2}2) =0.2f(0+\dfrac {0.2}2,\dfrac {1+0.1963}2)\\ =0.2\times \dfrac{1.0984^2-0.1^2}{1.0984^2+0.1^2}\\ =0.1967 \\ k_4=hf(x_0+h,y_0+k_3)=0.2\times …