Exam 10: First-Order Differential Equations

Solve the problem. -The system of equations $\frac { \mathrm { dx } } { \mathrm { dt } } = ( - 4 + 5 \mathrm { y } ) \mathrm { x }$ and $\frac { \mathrm { dy } } { \mathrm { dt } } = ( - 4 + \mathrm { x } ) \mathrm { y }$ 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). Find the equilibrium points.

Solve the problem. -A tank contains 100 gal of fresh water. A solution containing 2 lb/gal of soluble lawn fertilizer runs into the tank at the rate of 1 gal/min, and the mixture is pumped out of the tank at the rate of 2 gal/min. Find the Maximum amount of fertilizer in the tank and the time required to reach the maximum.

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

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. - $y ^ { \prime } = y - e ^ { x } - 1 , y ( 1 ) = 2 , d x = 0.5$

Sketch several solution curves. - $y ^ { \prime } = y - 2 \sqrt { y }$

Sketch several solution curves. - $\frac { d y } { d x } = y ^ { 3 } - 3 y$

Identify equilibrium values and determine which are stable and which are unstable. - $\frac { \mathrm { dy } } { \mathrm { dx } } = ( \mathrm { y } + 5 ) ( \mathrm { y } + 4 )$

Solve the differential equation. - $x y ^ { \prime } + 4 y = \frac { \cos x } { x ^ { 3 } } , x > 0$

Solve the problem. -A tank initially contains 120 gal of brine in which 40 lb of salt are dissolved. A brine containing 2 lb/gal of salt runs into the tank at the rate of 9 gal/min. The mixture is kept uniform by stirring and flows out of the tank at The rate of 6 gal/min. Write, in standard form, the differential equation that models the mixing process.

Solve. Round your results to four decimal places. -Use the Euler method with $\mathrm { dx } = 0.2$ to estimate $\mathrm { y } ( 3 )$ if $\mathrm { y } ^ { \prime } = - \mathrm { y }$ and $\mathrm { y } ( 2 ) = 2$ . What is the exact value of $\mathrm { y } ( 3 )$ ?

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

Sketch several solution curves. - $y ^ { \prime } = y ^ { 4 } - y ^ { 3 }$

Solve. Round your results to four decimal places. -Use the Euler method with $\mathrm { dx } = 0.5$ to estimate $\mathrm { y } ( 5 )$ if $\mathrm { y } ^ { \prime } = \mathrm { y } ^ { 2 } / \sqrt { 2 \mathrm { x } }$ and $\mathrm { y } ( 4 ) = - 3$ . What is the exact value of $\mathrm { y } ($ 5)?

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

Solve the initial value problem. - $2 \frac { d y } { d x } - 4 x y = 8 x ; y ( 0 ) = 23$

Solve. -A local pond can only hold up to 39 geese. Six geese are introduced into the pond. Assume that the rate of grow of the population is $\frac { \mathrm { dP } } { \mathrm { dt } } = ( 0.0013 ) ( 39 - \mathrm { P } ) \mathrm { P }$ where $t$ is time in weeks. Find a formula for the goose population in terms of $\mathrm { t }$ .

Solve the differential equation. - $x \frac { d y } { d x } + 2 y = 4 - \frac { 1 } { x } , x > 0$

Solve the differential equation. - $\frac { d y } { d x } - \frac { y } { x } = ( \ln x ) ^ { 5 }$

Solve the differential equation. - $x \frac { d y } { d x } = y + \left( x ^ { 2 } - 1 \right) ^ { 2 }$

Solve the initial value problem. - $\frac { d y } { d t } + 7 y = 3 ; y ( 0 ) = 1$