attached, just show me how they found the final answer by steps
PRACTICE PROBLEMS FOR EXAM 1 1. Find the solutions of y′ = (y + 1)2 satisfying the initial conditions (a) y(0) = 1; (b) y(0) = −1. What are the longest intervals of existence for each of these solutions ? 2. Find the general solutions of the following equations : (a) y′ + ty = t; (b) (t2 + t − 2)y′ − y = 0, t > 1; (c) ty′ − y = t3 cos t, t > 0, (d) (t + 1)y′ + ty = e−t, t > −1. 3. Use isoclines to plot solution curves for y′ = y2 − t. Do the solution curves ever intersect ? Explain. 4. The following questions concern the equation y′ = y2 − 2y. (a) Find the equilibrium solutions and sketch the phase diagram. (b) In the ty–plane, sketch solutions with the initial conditions y(0) = 2.1, y(0) = 1.9, y(0) = 1, and y(0) = −1. (c) If y(0) = 1.9, does the graph of y(t) have an inflection point ? If so, for what value of y ? 5. Use Euler’s method with step h = 0.1 to find a numerical approximation to x(0.5), the solution of x′ = −x + t with x(0) = 0. Compute approxima- tions to 3 decimal places. 6. Suppose a 20–gallon tank contains 10 gallons of clean water at time t = 0. Water containing 0.3 pounds per gallon of salt enters the tank at a rate of 2 gallons per minute. Suppose clean water enters the tank at a rate of 1 gallon per minute. Finally, suppose the liquid in the tank is well mixed at all times and saltwater exits the tank at a rate of 0.5 gallons per minute. Find the concentration of salt at the moment when the tank starts to overflow. 7. An apple pie, whose temperature was 220◦ F when removed from the oven, was set on a 40◦ F porch to cool. Some time later, its temperature was 180◦ F, and 15 minutes after that, it was 70◦ F. How long did it take the pie to cool down to 180◦ F ? 8. Solve the differential equation y′ = 1 + t e−y by using the substitution u = ey. 2 ANSWERS 1. (a) y = −1 + 1/(0.5− t), −∞ < t="">< 0.5;="" (b)="" y="−1," −∞="">< t=""><∞. 2. (a) y = 1+k e−t 2/2, (b) y = k ( t− 1 t + 2 )1/3 , (c) y = t2 sin t+t cos t+ct, (d) y = (ct + c − 1)e−t. 4. (a) y = 0 and y = 2; (c) yes, when y = 1. 5. 0.090. 6. (4− 24/5)/20 ≈ 0.113 7. about 2.45 minutes 8. y = ln (cet − t− 1). 2.="" (a)="" y="1+k" e−t="" 2/2,="" (b)="" y="k" (="" t−="" 1="" t="" +="" 2="" )1/3="" ,="" (c)="" y="t2" sin="" t+t="" cos="" t+ct,="" (d)="" y="(Ct" +="" c="" −="" 1)e−t.="" 4.="" (a)="" y="0" and="" y="2;" (c)="" yes,="" when="" y="1." 5.="" 0.090.="" 6.="" (4−="" 24/5)/20="" ≈="" 0.113="" 7.="" about="" 2.45="" minutes="" 8.="" y="ln" (cet="" −="" t−="">