Exam 10: First-Order Differential Equations

Solve the initial value problem. - $\theta ^ { 2 } \frac { d y } { d \theta } - 3 \theta y = \theta ^ { 5 } \sec \theta \tan \theta ; \theta > 0 , y ( \pi ) = 0$

Show that the curves are orthogonal. - $x ^ { 2 } + y ^ { 2 } = 5 \text { and } y ^ { 2 } = x ^ { 3 }$

Solve the initial value problem. - $y ^ { \prime } + y = 2 e ^ { x } ; y ( 0 ) = 22$

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 $x ^ { * }$ . -Using the given conditions, obtain a slope field and graph the particular solution over the specified interval. Then find the general solution of the differential equation. =y(2-y),y(0)= 0\leqx\leq4,0\leqy\leq3

Provide an appropriate response. -If a body of mass $m$ falling from rest under the action of gravity encounters an air resistance proportional to three times the square root of velocity, then the body's velocity $t$ seconds into the fall satisfies the equation: $\mathrm { m } \frac { \mathrm { dv } } { \mathrm { dt } } = \mathrm { mg } - 3 \mathrm { kv } ^ { 2 }$ where $\mathrm { k }$ is a constant that depends on the body's aerodynamic properties and the density of the air. Determine the equilibrium, velocity curve, and the terminal velocity for a $150 \mathrm { lb }$ skydiver $( \mathrm { mg } = 150 )$ with $\mathrm { k } = 0$ .

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 = -x(1 - y), y(2) = 3, dx = 0.2

Solve the problem. -The system of equations $\frac { \mathrm { dx } } { \mathrm { dt } } = ( - 5 + 6 \mathrm { y } ) \mathrm { x }$ and $\frac { \mathrm { dy } } { \mathrm { dt } } = ( - 5 + \mathrm { x } ) \mathrm { y }$ describes the growth rates of two symbiotic (dependent) species of animals (such as the rhinoceros and a type of bird which eats insects from its back). What happens to the rhinoceros population when the bird population decreases?

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 $x ^ { * }$ . -Using the given conditions, obtain a slope field and graph the particular solution over the specified interval. Then find the general solution of the differential equation. =y(3-y),y(0)= 0\leqx\leq5,0\leqy\leq4

Obtain a slope field and add to its graphs of the solution curves passing through the given points. - $y ^ { \prime } = y ( 1 - y ) \text { with } ( 0 , - 1 )$

Solve the problem. -If the switch is thrown open after the current in an RL circuit has built up to its steady-state value, the decaying current obeys the equation $\mathrm { L } \frac { \mathrm { di } } { \mathrm { dt } } + \mathrm { Ri } = 0$ . How long after the switch is thrown open will it take the current to fall to $35 \%$ of its original value?

Match the differential equation with the appropriate slope field. - $y ^ { \prime } = y ( y + 2 ) ( y - 2 )$

Solve. -A local pond can only hold up to 53 geese. Seven geese are introduced into the pond. Assume that the rate of grow th of the population is $\frac { \mathrm { dP } } { \mathrm { dt } } = ( 0.0015 ) ( 53 - \mathrm { P } ) \mathrm { P }$ where $t$ is time in weeks. How long will it take for the goose population to be 25 ?

Solve the differential equation. - $- x ^ { 3 } y ^ { \prime } + 2 x ^ { 2 } y = y ^ { 2 }$

Solve the problem. -A 100 gal tank is half full of distilled water. At time = 0, a solution containing 2 lb/gal of concentrate enters the tank at the rate of 4 gal/min, and the well-stirred mixture is withdrawn at the rate of 3 gal/min. When the tank Is full, how many pounds of concentrate will it contain?

Construct a phase line. Identify signs of y and y . - $\frac{d y}{d x}=y^{2}-9$

Solve the initial value problem. - $t \frac { d y } { d t } + 4 y = t ^ { 3 } ; t > 0 , y ( 2 ) = 1$

Identify equilibrium values and determine which are stable and which are unstable. - $\frac { d y } { d x } = y ^ { 2 } - 5 y$

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

Match the differential equation with the appropriate slope field. - $y ^ { \prime } = y + 2$

Match the differential equation with the appropriate slope field. - $y ^ { \prime } = x ^ { 2 } - y ^ { 2 }$