29.10. EXERCISES 577

71. Sometimes you have an equation of the form

y′′ = f(y,y′)

and you are looking for a function t → y(t) so the independent variable is missing.These can be massaged into a first order equation as follows. Let v = y′ and then youhave

v′ = f (y,v)

Now dvdt =

dvdy

dydt =

dvdy v. Thus we have

vdvdy

= f (y,v)

which is now a first order differential equation. Use this technique to solve the fol-lowing problems. This won’t always work. It is a gimmick which sometimes works.

(a) y′′+2y′ = 0,y(0) = 1,y′ (0) = 0

(b) y′′ = y′ (2y+1) ,y(0) = 0,y′ (0) = 1

(c) y′′ = 2yy′,y(0) = 0,y′ (0) = 1

(d) y′′ = y′(1−3y2

),y(0) = 1,y′ (0) = 0

(e) y′y′′ = 2,y(0) = 1,y′ (0) = 2

(f) y′′ = 2y,y(0) = 1,y′ (0) = 2

(g) y′y′′+3y = 0,y(0) = y′ (0) = 1

(h)(1+3t2

)y′′+6ty′− 3

t2 = 0,y′ (1) = 1,y(1) = 2. Hint: This is not like the abovebut d

dt

((1+3t2

)y′)

gives the first two terms.

(i) yy′′+(y′)2 = 0. Give a general solution involving two constants of integration.

(j) y′′+y(y′)2 = 0. Give a general solution involving two constants of integration.

(k) y′′y2−2y(y′)2 = 0, Give a general solution involving two constants of integra-tion.

(l) y′′y3−3y′y2 = 0,Give a general solution involving two constants of integration.

(m) 3(y′)2 y′′y2 + 2y(y′)4 = 0,Give a general solution involving two constants ofintegration.

72. Explain how you would proceed to solve an equation of the form y′′ = f (t,y′) wherethe function you are looking for is t→ y(t) . How many independent constants wouldyou have in a general solution?

Computer Algebra Problems

73. Give a graph of the solution to the following initial value problem on the interval[0,5]. y′ =−y3 +3y2 +2, y(0) = 0.

74. Give a graph of the solution to the following initial value problem on the interval[0,5].y′ =−y3 + xy2 +1, y(0) = 1.