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Exercises

I. Supply information to each DE in the indicated title:type (ode or pde), order(1st , 2nd, ), degree (1st,2nd, ), linearity (linear or nonlinear), dependent & independent variables (write the variable/spresent in the equation).

DE Type Order Degree LinearityDependentVariable(s)

IndependentVariable

1 y = 3x + e2x

2 tzdt

dz

dt

zd

2

3

3

4

3 zy

zx

x

z

2

2

2

2

4 1232

2

2

dx

dy

dx

ydey

5 0232

r

v

t

v

rt

v

6 wzdz

dwz

dz

wd54

3

3

sin

7 y 3xy + 4x2y y = ex + 2

8 032

2

3

32 ty

dt

ydt

dt

ydt cossin

9 122

2

xdx

dy

dx

ydey

10 tdtdyy

dtdy 35

II. Verify whether each given function is an explicit solution to the DE.1. y = x2 + 3x 1; x2y xy + y = x2 12. y = 4sin x 2x + 2; y + y + y = 4cosx 2x3. y = e3x 3cosx + 4x; y 3y + 3y 3y = 4 12x4. y1 = 2ex , y2 = 4ex , y3 = 31 ex; y + y = 05. y1 =

x

1, y2 =

x

2for all x 0, y3 =

2

1

x

for all x 2; y + y2 = 0

6. x1(t) = 2sin 2t, x2(t) = cos 2t, x3(t) = 3 cos 2t; 042

2

xdt

xd

7. y1 = ex x 1, y2 = cos x + sin x + ex x 1; y y + y y = x8.

214

x

xy

ln ,

226

xy ; x4y + 4x3y + 4x2y + 4 = 0, (x > 0)

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III. Determine whether the given function is the implicit solution of the given DE.1. x2y + 6 = 0, x2y + 4xy + 2y = 02. yex + xy = 1, y(x + ex) + 2y(1 + ex) + yex = 03. t2z 4 ln t = 0, 045

2

22 z

dt

dzt

dt

zdt

4. 3x2 + 5y2 +3x 10y 15 = 0, 3yy + y(y 1) =05. rsin + cos 2r+ 5 = 0, 0122

2

23

sinsinsincossin rd

dr

d

rd

d

rd

IV. Show that the given function is the general solution of the DE.1. y = Cex, y + y = 02.

kxy

1, y + y2 = 0

3. (y A)2 =Ax, 4x(y)2 + 2xy y = 04. y = C1 sin 2x + C2 cos 2x, y + 4y = 05. y =Aet + Be2t + t, 3223

2

2

xydt

dy

dt

yd

6. y = C1 sin x + C2x , (1 x cot x)y xy + y = 07. x = C1e2v + C2e2v + C3e4v , 01644

2

2

3

3

ydv

dx

dv

xd

dv

xd

8. y =Aex+ Bxex +Cex +2x2ex, y y y + y = 8ex