Exam 10: First-Order Differential Equations

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. - $y ^ { \prime } = 4 x e ^ { x ^ { 4 } } , y ( 1 ) = 3 , d x = 0.1$

Solve the problem. -An office contains $1000 \mathrm { ft } ^ { 3 }$ of air initially free of carbon monoxide. Starting at time $= 0$ , cigarette smoke containing $4 \%$ carbon monoxide is blown into the room at the rate of $0.5 \mathrm { ft } 3 / \mathrm { min }$ . A ceiling fan keeps the air in the room well circulated and the air leaves the room at the same rate of $0.5 \mathrm { ft } ^ { 3 } / \mathrm { min } ^ { \text {. Find } }$ the time when the concentration of carbon monoxide reaches $0.01 \%$ .

Solve the initial value problem. - $\frac { d y } { d x } + x y = 4 x ; y ( 0 ) = - 4$

Find the orthogonal trajectories of the family of curves. Sketch several members of each family. -y = -mx

Solve the problem. -A 200 gal tank is half full of distilled water. At time = 0, a solution containing 1 lb/gal of concentrate enters the tank at the rate of 4 gal/min, and the well-stirred mixture is withdrawn at the rate of 2 gal/min. When the tank Is full, how many pounds of concentrate will it contain?

Identify equilibrium values and determine which are stable and which are unstable. - $\frac { \mathrm { dy } } { \mathrm { dx } } = \mathrm { y } ^ { 2 } - 16$

Solve. -Solve the initial value problem $\frac { \mathrm { dP } } { \mathrm { dt } } = 7 \mathrm { kP } ^ { 2 } , \mathrm { P } ( 0 ) = \mathrm { P } _ { 0 }$

Solve the problem. -dy/dt = ky + f(t) is a population model where y is the population at time t and f(t) is some function to describe the net effect on the population. Assume k = .02 and y = 10,000 when t = 0. Solve the differential equation of y When f(t) = -8t.

Identify equilibrium values and determine which are stable and which are unstable. -y = (y - 4)(y - 6)(y - 7)

Construct a phase line. Identify signs of y and y . - $\frac{d y}{d x}=(y+1)(y-2)$

Solve the problem. -dy/dt = ky + f(t) is a population model where y is the population at time t and f(t) is some function to describe the net effect on the population. Assume k = .02 and y = 10,000 when t = 0. Solve the differential equation of y When f(t) = 11t.

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

Solve the problem. -The system of equations $\frac { d x } { d t } = ( - 2 + 3 y ) x$ and $\frac { d y } { d t } = ( - 2 + x ) y$ 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 is necessarily true of the two populations at the equilibrium points?

Determine which of the following equations is correct. - $\frac { 1 } { \sin x } \int \sin x d x =$

Solve the problem. -How many seconds after the switch in an RL circuit is closed will it take the current i to reach 20% of its steady state value? Express answer in terms of R and L and round coefficient to the nearest hundredth.

Sketch several solution curves. - $\frac { d y } { d x } = y ^ { 2 } - 1$

Obtain a slope field and add to its graphs of the solution curves passing through the given points. - $y ^ { \prime } = 3 ( y - 1 ) \text { with } ( 2,0 )$

Solve the differential equation. - $e ^ { x } \frac { d y } { d x } + 3 e ^ { x } y = 2 , x > 0$

The autonomous differential equation represents a model for population growth. Use phase line analysis to sketch solution curves for P(t), selecting different starting values P(0). Which equilibria are stable, and which are unstable? - $\frac { \mathrm { dP } } { \mathrm { dt } } = \mathrm { P } ( \mathrm { P } - 5 )$

Obtain a slope field and add to its graphs of the solution curves passing through the given points. - $\mathrm { y } ^ { \prime } = \frac { 2 \mathrm { y } } { \mathrm { x } } \text { with } ( - 2,0 )$