Exam 10: First-Order Differential Equations

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use Euler's method with the specified step size to estimate the value of the solution at the given point $x$ . Find the value of the exact solution at $x ^ { * }$ . - $y ^ { \prime } = 3 y ^ { 2 } / \sqrt { 2 x } , y ( 1 ) = - 1 , d x = 0.5 , x ^ { * } = 5$

Solve the differential equation. - $y ^ { \prime } + y \tan x = \cos x , - \pi / 2 < x < \pi / 2$

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places. -y = 2xy - 2y, y(2) = 3, dx = 0.2

Match the differential equation with the appropriate slope field. - $\mathrm { y } ^ { \prime } = \mathrm { x } - \mathrm { y }$

Solve the initial value problem. - $( x + 3 ) \frac { d y } { d x } - 2 \left( x ^ { 2 } + 3 x \right) y = \frac { e ^ { x ^ { 2 } } } { x + 3 } ; x > - 3 , y ( 0 ) = 0$

Solve. -A 57-kg skateboarder on a 2-kg board starts coasting on level ground at 5 m/sec. Let k = 3.2 kg/sec. How long will it take the skater's speed to drop to 3 m/sec?

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use Euler's method with the specified step size to estimate the value of the solution at the given point $x$ . Find the value of the exact solution at $x ^ { * }$ . - $y ^ { \prime } = y - e ^ { - 3 x } , y ( 0 ) = 2 , d x = 1 / 3 , x ^ { * } = 2$

Solve the differential equation. - $y ^ { \prime } e ^ { 5 x } + 5 y e ^ { 5 x } = 4 x$

Find the orthogonal trajectories of the family of curves. Sketch several members of each family. - $k x ^ { 2 } + y ^ { 2 } = 1$

Solve the initial value problem. - $\theta \frac { \mathrm { dy } } { \mathrm { d } \theta } + \mathrm { y } = \cos \theta ; \theta > 0 , \mathrm { y } ( \pi ) = 1$