# Solution of Differential Equations

**Runge-Kutta method***k**y*corresponding to the increment*h*of*x*.*f*(*x*,*y*) ;*y*(*x*) =_{0}*y*_{0}_{Calculation is done as follows :}*k*_{1}=*h f*(*x*,_{0}*y*),_{0}*k*_{2}=_{k3 = , k4 = hf (x0 + h, y0 + k3)}_{k = (k1 + 2k2 + 2k3 + k4) = (k1 + k4) + (k2 + k3)}_{The approximate value, y1 = y+ k}**Taylorâ€™s series method***y*â€² =*f*(*x*,*y*) and*f**x*_{0}) =*y*_{0}_{By taylorâ€™s theorem the series about a point x = x0}_{y = y0 + (x â€“ x0) (yâ€²)0 + }_{From this equation we can find y1 of y for x = x1 and then yâ€², yâ€³, yâ€³â€² are found out.}**Eulerâ€™s method***y*â€² =*f*(*x*,*y*) and*y*(*x*) =_{0}*y*_{0}_{when h â†’ 0, by Taylorâ€™s series}_{y(x + b) = y (x) + h yâ€²(x) = y(x) + h f (x, y)}_{Thus, yn+1 = yn h f( xn, yn)}_{where, h = i.e. (xn = x0 + nh)}