Exam 10: Differential Equations

Use Euler's method with n = 5 on the interval 0 ≤ t ≤ $\frac { 1 } { 2 }$ to approximate the solution $f ( t ) \text { to } y ^ { \prime } = - ( y + t ) , y ( 0 ) = - 1 \text {. }$ Is the following the correct answer? =0; =-1 =0.1; =-0.9 =0.2; =-0.82 =0.3; =-0.758 =0.4; =-0.7122 =0.5; =-0.68098

Consider the differential equation $y ^ { \prime }$ = g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false. -Let $y ^ { \prime }$ = $y ^ { 3 }$ . Which of the following properties hold for the solution y = f(t) determined by the initial condition y(0) = -2? (I) It is always increasing. (II) It has an inflection point. (III) It is always concave down.

Use Euler's method with n = 4 to approximate the solution f(t) to $y ^ { \prime }$ = t + y - 1, $y ( 0 ) = 2$ for $0 \leq t \leq 2$ . Estimate f(2). Enter just a reduced fraction of form $\frac { a } { b }$ .

Let f(t) be the solution of $y ^ { \prime }$ = ty + 0.2, y(0) = 3. Use Euler's method on $0 \leq t \leq 1$ with n = 2 to estimate f(1). Is $f ( 1 ) \approx 3.875$ the correct answer?

A certain chemical vaporizes when exposed to the air. Suppose f(t) is the amount of chemical present. It is found that the rate of vaporization of the chemical is proportional to the amount of chemical present squared. Write a differential equation satisfied by f(t) . Does this equation accurately represent this situation: $\mathrm { y } ^ { \prime } = \mathrm { ky } 2 ; \mathrm { k } < 0 \text { ? }$

Suppose that an epidemic is spreading at a rate proportional to the square of the infected population. Let f(t) be the number of infected people at time t, and suppose y = f(t) satisfies a differential equation $y ^ { \prime } = g ( y )$ . Which of the following sets of curves could represent solutions of $y ^ { \prime } = g ( y ) ?$ [Hint: First determine the differential equation $y ^ { \prime }$ = g(y) .]