Exam 6: Slope Fields and Eulers Method

Find the particular solution of the differential equation $4 x y ^ { t } - y = x ^ { 3 } - 2 x \text { that }$ satisfies the boundary condition $y \left( \sqrt { \frac { 22 } { 3 } } \right) = 0$ .

$\text { The rate of change of } N \text { is proportional to } N \text {. When } t = 0 , N = 200 \text { and when }$ $t = 1 , N = 360$ . What is the value of $N$ when $t = 4$ ? Round your answer to three decimal places.

Use Euler's Method to make a table of values for the approximate solution of the following differential equation with specified initial value. Use 5 steps of size 0.15. $y^{t}=4 x+2 y, \quad y(0)=2$

$\text { Which of the following is a solution of the differential equation } 7 y ^ { \prime \prime } + 7 y = 0 \text { ? }$

Solve the differential equation. $\frac { d y } { d x } = x + 8$

$\text { At time } t = 0 \text { minutes, the temperature of an object is } 160 ^ { \circ } \mathrm { F } \text {. The temperature of the }$ object is changing at the rate given by the differential equation $\frac { d y } { d t } = - \frac { 1 } { 2 } ( y - 92 )$ . Use Euler's Method to approximate the particular solutions of this differential equation at $t = 2$ . Use a step size of $h = 0.1$ . Round your answer to one decimal place.

$\text { The half-life of the radium isotope Ra-226 is approximately } 1,599 \text { years. If the initial }$ quantity of the isotope is $38 \mathrm {~g}$ , what is the amount left after 1,000 years? Round your answer to two decimal places.

Suppose an eight-pound object is dropped from a height of 5000 feet, where the air resistance is proportional to the velocity. Write the velocity as a function of time if its velocity after 7 Seconds is approximately -75 feet per second. Use a graphing utility or a computer algebra system. Round numerical answers in your answer to four places.

Find the particular solution of the differential equation $y ^ { t } + ( 8 x - 20 ) y = 0 \text { that }$ satisfies the boundary condition $y ( 5 ) = 2$ .

$\text { Find the particular solution of the differential equation } x ^ { 9 } y ^ { t} + 8 y = e ^ { 1 / x ^ { 8 } } \text { that }$ satisfies the initial condition $y ( 1 ) = e$ .

A 200-gallon tank is full of a solution containing 25 pounds of concentrate. Starting at time $t = 0$ , distilled water is added to the tank at a rate of 10 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the time at which the amount of concentrate in the tank reaches 15 pounds. Round your answer to one decimal place.

$\text { Find the function } y = f ( t ) \text { passing through the point } ( 0,12 ) \text { with the first derivative }$ $\frac { d y } { d t } = \frac { 6 } { 7 } y$

Solve the differential equation. $y ^ { t } = \frac { - 2 x } { y }$

Suppose that the population (in millions) of Paraguay in 2007 was 6.7 and that the expected continuous annual rate of change of the population is $0.024$ . Find the exponential growth model $P = C e ^ { k t }$ for the population by letting $t = 0$ correspond to 2000 . Round your answer to four decimal places.

Find an equation of the graph that passes through the point (7, 3) and has the slope $y ^ { t } = \frac { 5 y } { 2 x }$

Find the logistic equation that satisfies the following differential equation and initial condition. $\frac { d y } { d t } = 3.4 y \left( 1 - \frac { y } { 53 } \right) , y ( 0 ) = 13$

$\text { Find the particular solution of the differential equation } \frac { d y } { d x } + 7 x ^ { 3 } y = x ^ { 3 } \text { passing }$ through the point $\left( 0 , \frac { 3 } { 2 } \right)$ .

$\text { Use } u ( x , y ) = x ^ { 3 } y \text { as a integrating factor to find the general solution of the }$ differential equation $\left( 4 y ^ { 3 } + 7 x ^ { 3 } y \right) d x + \left( 4 x y ^ { 2 } + 2 x ^ { 4 } \right) d y = 0$

$\text { The logistic function } P ( t ) = \frac { 24 } { 1 + 3 e ^ { - 2 t } } \text { models the growth of a population. Identify }$ the initial population.

Use integration to find a general solution of the differential equation. $\frac { d y } { d x } = 5 x ^ { 3 } + x$