1. Find the order of the equation.
a. d2ydx2+2dydx d3ydx3+x=0
b. y″−3y′+2y=x7
c. y′−y7=0
d. y″y−(y′)2=2
2. Verify that the function is a solution of the differential equation on some interval, for any choice of the arbitrary constants appearing in the function.
a. y=ce2x;y′=2y
b. y=x23+cx;xy′+y=x2
c. y=12+ce−x2;y′+2xy=x
d. y=(1+ce−x2/2)(1−ce−x2/2)−1;2y′+x(y2−1)=0
e. y=tan(x33+c);y′=x2(1+y2)
f. y=(c1+c2x)ex+sinx+x2;y″−2y′+y=−2cosx+x2−4x+2
g. y=c1ex+c2x+2x;(1−x)y″+xy′−y=4(1−x−x2)x−3
h. y=x−1/2(c1sinx+c2cosx)+4x+8; x2y″+xy′+(x2−14)y=4x3+8x2+3x−2
3. Find all solutions of the equation.
a. y′=−x
b. y′=−xsinx
c. y′=xlnx
d. y″=xcosx
e. y″=2xex
f. y″=2x+sinx+ex
g. y‴=−cosx
h. y‴=−x2+ex
i. y‴=7e4x
4. Solve the initial value problem.
a. y′=−xex,y(0)=1
b. y′=xsinx2,y(√π2)=1
c. y′=tanx,y(π/4)=3
d. y″=x4,y(2)=−1,y′(2)=−1
e. y″=xe2x,y(0)=7,y′(0)=1
f. y″=−xsinx,y(0)=1,y′(0)=−3
g. y‴=x2ex,y(0)=1,y′(0)=−2,y″(0)=3
h. y‴=2+sin2x,y(0)=1,y′(0)=−6,y″(0)=3
i. y‴=2x+1,y(2)=1,y′(2)=−4,y″(2)=7
5. Verify that the function is a solution of the initial value problem.
a. y=xcosx;y′=cosx−ytanx,y(π/4)=π4√2
b. y=1+2lnxx2+12;y′=x2−2x2y+2x3,y(1)=32
c. y=tan(x22);y′=x(1+y2),y(0)=0
d. y=2x−2;y′=−y(y+1)x,y(1)=−2
6. Verify that the function is a solution of the initial value problem.
a. y=x2(1+lnx);y″=3xy′−4yx2,y(e)=2e2,y′(e)=5e
b. y=x23+x−1;y″=x2−xy′+y+1x2,y(1)=13,y′(1)=53
c. y=(1+x2)−1/2;y″=(x2−1)y−x(x2+1)y′(x2+1)2,y(0)=1,y′(0)=0
d. y=x21−x;y″=2(x+y)(xy′−y)x3,y(1/2)=1/2,y′(1/2)=3
7. Suppose an object is launched from a point 320 feet above the earth with an initial velocity of 128 ft/sec upward, and the only force acting on it thereafter is gravity. Take g=32 ft/sec2.
- Find the highest altitude attained by the object.
- Determine how long it takes for the object to fall to the ground.
8. Let a be a nonzero real number.
- Verify that if c is an arbitrary constant then y=(x−c)a is a solution of y′=ay(a−1)/a on (c,∞).
- Suppose a<0 or a>1. Can you think of a solution of (B) that isn’t of the form (A)?
9. Verify that y={ex−1,x≥0,1−e−x,x<0,
is a solution of
y′=|y|+1 on (−∞,∞).
10.
(a) Verify that if c is any real number then y=c2+cx+2c+1 satisfies y′=−(x+2)+√x2+4x+4y2 on some open interval. Identify the open interval.
(b) Verify that y1=−x(x+4)4 also satisfies (B) on some open interval, and identify the open interval. (Note that y1 can’t be obtained by selecting a value of c in (A).)