Suppose $y ^ { \prime } = 2 y - x$ and $y ( 0 ) = 2$ .Estimate $y ( 0.05 )$ and $y ( 0.1 )$ using the slope given by the differential equation.
A)4 and 3.9
B)5.95 and 17.75
C)3.95 and 3.9
D)7.95 and 17.95
E)2.2 and 2.4175

Question

Quizplus · Accepted Answer

To estimate $y(0.05)$ and $y(0.1)$ using the slope given by the differential equation $y' = 2y - x$, we use Euler's method with a step size of $0.05$. For $y(0.05)$, we start with $y(0) = 2$ and $x = 0$. The slope at $x = 0$ is $y' = 2(2) - 0 = 4$. So, $y(0.05) \approx y(0) + 0.05 \cdot y' = 2 + 0.05 \cdot 4 = 2.2$.For $y(0.1)$, we use the value of $y(0.05) = 2.2$ and $x = 0.05$. The slope at $x = 0.05$ is $y' = 2(2.2) - 0.05 = 4.35$. So, $y(0.1) \approx y(0.05) + 0.05 \cdot y' = 2.2 + 0.05 \cdot 4.35 = 2.4175$.Therefore, the estimates are $y(0.05) = 2.2$ and $y(0.1) = 2.4175$.

Suppose y′=2y−xy ^ { \prime } = 2 y - xy′=2y−x

Suppose $y ^ { \prime } = 2 y - x$