## Calculus with Applications (10th Edition)

The actual result: $3.118$ The approximation: $3.139$ Difference : $3.139-3.118=0.021$
We are given $\frac{dy}{dx}=\frac{2x}{y}$ Using Euler's method $g(x,y)=\frac{2x}{y}$ Since $x=0, y=3$ $g(x_{0},y_{0})=0$ and $y_{1}=y_{0}+g(x_{0},y_{0})h=3+0\times0.1=3$ Now $x_{1}=0.1, y_{1}=3$ and $g(x_{1},y_{1})=0.067$ Then $y_{2}=3+(0.067)\times0.1=3.0067$ $y_{3}=3.0067+(0.133)\times0.1=3.02$ $y_{4}=3.02+(0.199)\times0.1=3.04$ $y_{5}=3.04+(0.263)\times0.1=3.07$ $y_{6}=3.07+(0.39)\times0.1=3.1$ $y_{7}=3.1+(0.387)\times0.1=3.139$ $y(0.4)=3.139$ $\frac{dy}{dx}=\frac{2x}{y}$ $\int ydy=\int xdx$ $\frac{1}{2}y^{2}=x^{2} + C$ With $x=0, y=3$ $C=\frac{9}{2}$ $\frac{1}{2}y^{2}=x^{2} + \frac{9}{2}$ $y(0.6) \rightarrow \frac{1}{2}y^{2}=(0.6)^{2} + \frac{9}{2}$ $\rightarrow y=\frac{9\sqrt 3}{5}\approx 3.118$ Difference between the result and approximation: $3.139-3.118=0.021$