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#### karush

##### Well-known member

- Jan 31, 2012

- 2,886

$\tiny{2.7.2}$

(a) Find approximate values of the solution of the given initial value problem\\

at $t = 0.1, 0.2, 0.3$, and $0.4$ using the Euler method with $h = 0.1$.

(b) Repeat part (a) with h = 0.05. Compare the results with those found in (a).

(c) Repeat part (a) with h = 0.025. Compare the results with those found in (a) and (b).

(d) Find the solution $y=\phi(t)$ of the given problem and evaluate

$\phi(t)$ at $t = 0.1,\quad 0.2, \quad 0.3,$ and $0.4$.

#1. $\quad\displaystyle

y'=3+t-y \quad y(0)=1$

ok assume first step is to get a general solution

rewrite

$y'+y=3+t $

then

$ey'+ey=(ey)'=3+t$

so

$\displaystyle ey=\int{(3+t)} \, dt= \frac{t^2}{2} + 3 t + c$

isolate

$\displaystyle y=\frac{t^2}{2e} + 3 te^{-1} + c^{-e}$

$\color{red}{(a) 1.2, 1.39, 1.571, 1.7439}$

$\color{red}{(b) 1.1975, 1.38549, 1.56491, 1.73658}$

$\color{red}{(c) 1.19631, 1.38335, 1.56200, 1.73308}$

$\color{red}{(d) 1.19516, 1.38127, 1.55918, 1.72968}$

Red is book answer

If I can get #1 probably 2,3 and 4 will be a slide

which are

2. $\quad y'=2y-1 \quad y(0)=1$

3. $\quad\displaystyle

y'=y'=0.5-t+2y, \quad y(0)=1$

4. $\quad\displaystyle

3\cos{t} -2y \quad y(0)=0 $

(a) Find approximate values of the solution of the given initial value problem\\

at $t = 0.1, 0.2, 0.3$, and $0.4$ using the Euler method with $h = 0.1$.

(b) Repeat part (a) with h = 0.05. Compare the results with those found in (a).

(c) Repeat part (a) with h = 0.025. Compare the results with those found in (a) and (b).

(d) Find the solution $y=\phi(t)$ of the given problem and evaluate

$\phi(t)$ at $t = 0.1,\quad 0.2, \quad 0.3,$ and $0.4$.

#1. $\quad\displaystyle

y'=3+t-y \quad y(0)=1$

ok assume first step is to get a general solution

rewrite

$y'+y=3+t $

then

$ey'+ey=(ey)'=3+t$

so

$\displaystyle ey=\int{(3+t)} \, dt= \frac{t^2}{2} + 3 t + c$

isolate

$\displaystyle y=\frac{t^2}{2e} + 3 te^{-1} + c^{-e}$

$\color{red}{(a) 1.2, 1.39, 1.571, 1.7439}$

$\color{red}{(b) 1.1975, 1.38549, 1.56491, 1.73658}$

$\color{red}{(c) 1.19631, 1.38335, 1.56200, 1.73308}$

$\color{red}{(d) 1.19516, 1.38127, 1.55918, 1.72968}$

Red is book answer

If I can get #1 probably 2,3 and 4 will be a slide

which are

2. $\quad y'=2y-1 \quad y(0)=1$

3. $\quad\displaystyle

y'=y'=0.5-t+2y, \quad y(0)=1$

4. $\quad\displaystyle

3\cos{t} -2y \quad y(0)=0 $

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