Euler's methods for differential equations Quiz

1. $\frac{dy}{dx}=2x+y$, $y(0)=1$, $h=0.5$. Find $y_1$ using Euler's method.

• A. $1.5$
• B. $2$
• C. $1$
• D. $2.5$

2. $\frac{dy}{dx}=y$, $y(0)=2$, $h=0.1$. Find $y(0.2)$ by Euler's method.

• A. $2.2$
• B. $2.4$
• C. $2.42$
• D. $2.44$

3. In the improved Euler formula $y_{r+1}=y_{r-1}+2hf(x_r,y_r)$, which gradient is used?

• A. At $x_{r-1}$
• B. At $x_{r+1}$
• C. At the end of the interval
• D. At $x_r$, the midpoint

4. $\frac{dy}{dx}=x^2$, $y(0)=0$, $h=1$. Euler's estimate of $y(2)$ is:

• A. $1$
• B. $0$
• C. $5$
• D. $8$

5. The solution curve is concave ($y''<0$). Euler's method gives:

• A. an exact value
• B. an overestimate
• C. an underestimate
• D. a random error