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1.2E: Basic Concepts (Exercises)

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1. Find the order of the equation.

a. d2ydx2+2dydx d3ydx3+x=0

b. y3y+2y=x7

c. yy7=0

d. yy(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+cex2;y+2xy=x

d. y=(1+cex2/2)(1cex2/2)1;2y+x(y21)=0

e. y=tan(x33+c);y=x2(1+y2)

f. y=(c1+c2x)ex+sinx+x2;y2y+y=2cosx+x24x+2

g. y=c1ex+c2x+2x;(1x)y+xyy=4(1xx2)x3

h. y=x1/2(c1sinx+c2cosx)+4x+8; x2y+xy+(x214)y=4x3+8x2+3x2

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=cosxytanx,y(π/4)=π42

b. y=1+2lnxx2+12;y=x22x2y+2x3,y(1)=32

c. y=tan(x22);y=x(1+y2),y(0)=0

d. y=2x2;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=3xy4yx2,y(e)=2e2,y(e)=5e

b. y=x23+x1;y=x2xy+y+1x2,y(1)=13,y(1)=53

c. y=(1+x2)1/2;y=(x21)yx(x2+1)y(x2+1)2,y(0)=1,y(0)=0

d. y=x21x;y=2(x+y)(xyy)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.

  1. Find the highest altitude attained by the object.
  2. Determine how long it takes for the object to fall to the ground.

8. Let a be a nonzero real number.

  1. Verify that if c is an arbitrary constant then y=(xc)a is a solution of y=ay(a1)/a on (c,).
  2. 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={ex1,x0,1ex,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).)


This page titled 1.2E: Basic Concepts (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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