Calculus B for Economics
Exercise Number 1
1) Find the domain of the following functions. For a function of two variables only, sketch
the domain.
a) f (x, y) =
d) z =
√
√
1
x2 + y 2
1
1
1
e) u = √ + √ + √
y
x
z
xy
x+y+
c) z = ln(y 2 − 4x + 8)
b) z =
√
x−y
f) u =
p
R 2 − x2 − y 2 − z 2
2) Verify that the function F (x, y) = xy satisfies the identity
F (ax + bu, cy + dv) = acF (x, y) + bcF (u, y) + adF (x, v) + dbF (u, v)
3) Determine the boundary of the following sets. Determine if the set is open or closed.
Draw the set.
a) {(x, y) : 2 ≤ x ≤ 4 , 1 ≤ y ≤ 3}.
b) {(x, y) : 1 < x2 + y 2 < 4}.
c) {(x, y) : 1 < x2 ≤ 4}.
d) {(x, y) : y < x2 }.
4) Given that the following limits exist, compute them.
a)
lim
(x,y)→(0,0)
p
−
x2 + y 2
x2 + y 2 + 1 − 1
5) Prove that the limit lim(x,y)→(0,0)
x+y
x−y
1
e x2 +y2
b)
lim
(x,y)→(0,0) x4 + y 4
does not exist.
6) Find the domain where the following functions are not continuous
a) z =
2
2
x + y2
b) z =
1
x−y
c) z =
y 2 + 2x
y 2 − 2x
7) Compute the first partial derivatives of the following functions
a) z = xyln(x + y)
y
c) w = xyz + yzv + zvx + vxy
b) u = x z
8) For the function f (x, y) = x3 y − y 3 x compute
∂f
+ ∂f
∂x
∂y
∂f ∂f
∂x ∂y
1
at the point (x, y) = (1, 2).
9) Prove that u =
x2 y 2
x+y
satisfies the identity x ∂u
+ y ∂u
= 3u.
∂x
∂y
10) Let V denote the volume of a truncated cone with radius bases r and R, and with
height h. Thus V = 13 πh(R2 + Rr + r2 ). Find the rate of change of V with respect to each
of the variables.
2
Calculus B for Economics
Exercise Number 2
1) Compute all second order partial derivatives of the following functions
a) z = y lnx
b) u = x2 + y 2 + z 2
p
∂2u
2) For the function u = x2 + y 2 + z 2 − 2xz compute ∂y∂z
.
3
∂ z
3) For the function z = ln(x2 + y 2 ) compute ∂x∂y
2.
∂6v
m n p
4) For the function v = x y z compute ∂x∂y3 ∂z2 .
5) Prove that the function u = ln √ 21 2 satisfies the identity
x +y
3
∂2u
∂x2
+
∂2u
∂y 2
= 0.
2
∂ u
∂ u
6) Prove that the function u = exyz satisfies the identity ∂x∂y∂z
= xy ∂x∂y
+ 2x ∂u
+ u.
∂x
7) Write down the differential of the following functions
√
x
x3 + 1
a) y =
b) y = 3
a+b
x −1
√
1
2 3
c) y = (1 + x − x )
d) y = 3− x2 + 3x3 − 4 x
8) Using linearization, approximate the following numbers
1
a) 1010 3
1
b) √
22
3
c) 33 5
d) e0.2
e) ln1.1
9) To construct a cube whose volume is 1000 m3 , use differentials to determine the
allowed mistake on the edge of the cube, so that the volume will be in the range of ±3 m3
from the planed volume.
10) For the following functions, write down the Taylor polynomial of order n around
a=0
a) e−x
11) For the function
b)
1
,
1+x
x
1
1+x
c) ln(1 + x)
write down the Taylor polynomial of order 4 around a = 1.
12) For the function e , write down the Taylor polynomial of order 4 around a = 1.
13) Using a suitable Taylor polynomial of order 6, estimate the value of e0.5 . Give a
bound for the mistake.
14) Using a suitable Taylor polynomial of order 3, estimate the value of ln1.3. Give a
bound for the mistake.
1
Calculus B for Economics
Exercise Number 3
1) Write down the linearization of the following functions at the point (a, b)
a) z = x2 y −2
b) z = ex−y − x2 y
c) u = xy 2 e−z
d) u = x2 y + y 2 z + z 2 x
2) Using a suitable function of two variables, use linearization to approximate the numbers
1
a) 1.042.02
1
b) ln(1.03 3 + 0.98 4 − 1)
3) Write down the differential of the following functions
a) z = x2 y −2
b) z = ex−y − x2 y
c) u = xy 2 e−z
d) u = x2 y + y 2 z + z 2 x
4) Using the chain rule compute
dz
dt
a) z = x2 + xy + y 2 where x = t2 and y = t.
b) z = ln(x2 + y 2 ) where x = et and y = lnt.
5) Let z =
x2
y
where x = u − 2v and y = v + 2u. Using the chain rule, compute
∂z
∂u
and
∂z
.
∂v
∂z
∂z
+ x ∂y
= x.
6) Let z = y + F (u) where u = x2 − y 2 . Using the chain rule prove that y ∂x
2y
∂2y
∂
7) If y = f (x − at) + g(x + at) prove that ∂t2 = a2 ∂x2 .
8) If z = yf (x2 − y 2 ) prove that
1 ∂z
x ∂x
+
1 ∂z
y ∂y
=
z
.
y2
9) Check that the following functions are homogenous, and verify Euler Theorem
x
a) z = x3 + xy 2 − 2y 3
b) z = e y
c) f (x, y, z) = xyz − 2x3 + 5yz 2
d) z =
x3
1
− y3
(1)
10) Suppose that the radius of the upper base of a truncated cone is 10 cm, the radius of
the lower base is 12 cm, and the height is 18 cm. What is the rate of change of the volume of
1
the cone if the radius of the upper base is decreasing at a rate of 2 cm per second, the radius
of the lower base is increasing at the rate of 3 cm per second, and the height is decreasing
at the rate of 4 cm per second?
11) The height of a cone is 20 cm. The radius of the base is 14 cm. What is the rate of
change of the volume of the cone, if the height is decreasing at the rate of 2 cm per second,
and the radius is increasing at the rate of 3 cm per second?
12) Using a suitable function of two variables, compute y 0 for the following implicit
functions
a) x3 + y 3 − 3axy = 0
x+y
c)
= y 2 e−y
x
b) ln(xy + y 2 ) − ex+y = 0
2
d) y − xx = 0
13) Find the equation of the tangent line to the ellipse
x2
2
+
y2
4
= 1 at the point (1,
14) Find the equation of the tangent line to the function xy − a = 0 at (a, 1).
2
√
2).
Calculus B for Economics
Exercise Number 4
1) Write down the Taylor expansion of order two around the point (a, b)
for the following functions
a) z = ln[(1 + x)(1 + y)] around (a, b) = (0, 0).
√
b) z = x + y around (a, b) = (1, 0).
c) z = x2 + xy around (a, b) = (2, 3).
d) z = ex ln(1 + y) around (a, b) = (0, 0).
2) Use the Taylor expansion of order two of a suitable function, in order
to approximate the number e0.1 ln(0.99). Compare this approximation with
the linear approximation.
3) Using the definition only, compute the directional derivative of
a) f (x, y) = 2x2 + y 2 at the point (−1, 1) in the direction of the unit vector
u = ( 35 , − 54 ).
b) f (x, y, z) = xy + yz + zx at the point (1, −1, 2) in the direction of the unit
vector u = ( 73 , 67 , − 27 ).
4) Compute the gradient of the functions
a) z = x2 y −2
b) z = ex−y − x2 y
c) u = xy 2 e−z
d) u = x2 y + y 2 z + z 2 x
5) Compute the gradient of the following functions at the given points
a) z =
2x
x−y
2
at the point (3, 1).
b) z = 2x − 3xy + 4y 2 at the point (2, 3).
6) By first computing the gradient, compute the directional derivatives
of the functions given in exercise 3).
1
Review Exercises
1) Let U and V be two open sets in the plane. Prove that U ∩ V is an
open set in the plane. Is the union of U and V an open set? Prove your
answer.
2) A function u(x, y) is said to be harmonic if uxx + uyy = 0. Similarly,
the function u(x, y, z) is said to be harmonic if uxx + uyy + uzz = 0.
p
a) Verify that the function z = ln x2 + y 2 is harmonic.
b) For what values of a, b, c, d, e, f the function u(x, y, z) = ax2 + by 2 + cz 2 +
dxy + exz + f yz is harmonic.
3) For what values of a and b the function z = (x + y)eax+by satisfy the
∂2z
∂z
∂z
equation ∂x∂y
− ∂x
− ∂y
+ z = 0.
4) Using the Taylor expansion of ex around a = 0, calculate the value of
e0.5 with an error less than 0.001.
5) Suppose that z = f (u, v) where u = xy and v = xy . Using the chain
∂z
∂z
rule, write down an expression for the partial derivatives ∂x
and ∂y
. Using
this, express
∂z
∂u
and
∂z
∂v
in terms of
∂z
∂x
and
∂z
.
∂y
6) As in one and two variables, we can write the linearization at (a, b, c)
of a function of three variables. It is given by
L(x, y, z) = f (a, b, c) + fx (a, b, c)(x − a) + fy (a, b, c)(x − b) + fz (a, b, c)(x − c)
For a suitable function, use this linearization to approximate the number
(1.98)3
p
(3.01)2 + (3.97)2
∂z
∂z
7) If yz 4 + x2 y 2 = exyz , find the partial derivatives ∂x
and ∂y
.
√
8) What is the maximum rate of change of the function f (x, y) = x2 y+ y
at the point (2, 1)? In what direction it occurs?
2
Calculus B for Economics
Exercise Number 5
1) For the following functions, find all critical points. If possible, determine if the point is a maximum or a minimum point.
a) z = 2x − x2 − y 2
b) z = x3 − 3x + y
x y
d) z = −
y x
p
√
f) z = y 1 + x + x 1 + y
c) z = |x| + |y|
e) z = e2x (x + y 2 + 2y)
2 −y 2
g) z = e−x
h) z = xy(a − x − y)
a 6= 0
i) u = 3lnx + 2lny + 5lnz + ln(22 − x − y − z) j) z = x3 y 2 (12 − x − y) x > 0; y > 0
k) z = xy + x2 + y 2 + x − y + 1
l) u = 2x2 + y 2 + 2z − xy − xz
m) z = x3 + y 3 − 3xy
2) Suppose that 0 < a < b. Find two numbers x and y, both between a
and b, which give a critical point for the function
xy
(a + x)(x + y)(b + y)
3) For any real number m, show that (0, 0) is a critical point for the
function f (x, y) = x2 + y 2 − mxy. Is this a minimum or a maximum point
for f (x, y)? Explain.
4) For the following functions, find the extreme points subject to the
given constraints
1
a) z = x2 + y 2
xy = 1.
b) z = xy 2
x2 + y 2 = 1.
c) z = x + y
x4 + y 4 = 1.
d) z = xm + y m
1 1
e) z = +
x y
g) z = xy
x + y = 2 (x, y ≥ 0, m > 0).
1
1
1
+
=
a 6= 0.
x2 y 2
a2
1 1 1
+ + = 1.
x y z
x2 + y 2 = 2a2 a 6= 0
h) u = xyz
x+y+z =5
i) u = x + y + z
x2 + y 2 + z 2 = 1
f) u = x + y + z
2
xy + xz + yz = 8.
x − y = 1.
Calculus B for Economics
Exercise Number 6
In each of the following exercises 1)- 8), find the absolute maximum and
the absolute minimum of the given function in the given domain. Draw the
domain.
1) z = x2 − y 2 in the domain x2 + y 2 ≤ 4.
2) z = x2 + 2xy − 4x + 8y in the domain closed by x = 0, y = 0, x = 1
and y = 2.
3) z = x2 y(4 − x − y) in the domain closed by x = 0, y = 0 and x + y = 6.
2
2
4) z = e−x −y (2x2 + 3y 2 ) in the domain x2 + y 2 ≤ 4.
5) z = 2x2 − 4x + y 2 − 4y + 1 in the domain closed by x ≥ 0, y = 2 and
y = 2x.
6) z = x2 + xy + y 2 − 6x in the domain closed by 0 ≤ x ≤ 5 and
−3 ≤ y ≤ 3.
7) z =
8) z =
x+3y
in the domain closed by y = x2 and
x
1
in the domain 0 < x2 + y 2 ≤ 1.
x2 +y 2
y = −x2 + 2.
9) Among all rectangles whose diagonal is p find the one with maximal
area.
10) Find the distance from the point (p, 4p) to the parabola y 2 = 2px.
11) Find the size of a box whose volume is maximal, if the sum of the
areas of all its sizes is 6a2 .
12) Among all triangulares with a given perimeter p, which one has the
p
maximal area? Use the identity S = p(p − a)(p − b)(p − c) where p =
1
(a
2
+ b + c) and a, b and c are the edges of the triangular.
13) Let C denote the intersection of the plane x + y + z = 1 with the
1
cylinder x2 + y 2 = 2. Find the points on C whose height above the x, y plane
is maximal. Similar for minimal.
14) The purpose of this exercise is to show that the conditions given by
Lagrange multipliers are necessary but not sufficient.
Show using Lagrange multipliers that the points (4, 4) and (−4, −4) are
critical points for the function z = x + y subject to the constraint xy = 16.
By substituting the constraint into the function show that z = x + y has no
extreme points subject to the constraint xy = 16.
2
Calculus B for Economics
Exercise Number 7
1) A company uses three type of materials with quantities x, y, z to produce a certain product. The profit of the company is given by the function
f (x, y, z) = 50x2/5 y 1/5 z 1/5 . If one quantity of the first type costs 80, the
second 12 and the third 10, and if the budget of the company is 24000, what
is the combination that will give the maximal profit?
2) A company produces x units of one product and y units of another
product. The expenses the company has is given by the function f (x, y) =
5x2 + 2xy + 3y 2 + 800.
a) If the company produces 39 products what is the combination that will
minimize the expenses.
b) Give an estimation for the change in the expenses, if the company decides
to produce 40 products. The same if it decides to produce 38 products.
3) The production of q quantities of a certain product, depends on the
number w of people working on the product and k, the amount of money
invested in the product. This function is given by q = 6w3/4 k 1/4 . The cost
of labor is 10 per person and the cost on money is 20 per unit. The total
budget is 3000.
a) What is the optimal number of people and units of money needed in order
the produce the largest number of products.
b) Show that at the point you found in part a), the ratio of
∂q
∂w
and
∂q
∂k
is
equal to the ratio between the cost of one unit of labor and the cost of one
unit of money.
c) Using part a) estimate what happens if the budget is increased by one.
1
4) a) A company plans to sell a product at a price of 150. It estimates
that if it will invest x thousands on development and y thousands on advertisement it will sell
320y
y+2
+
160x
x+4
units. A production of one unit costs 50. If
the company have 8000 to invest in development and on advertisement, how
should the money be divided in order to ensure the maximal profit?
b) If the company decides to spend 8100 instead 8000, use part a) to estimate
how this will change its maximal profit.
c) If the company has an unlimited budget, how much should it spend on
development and how much it should spend on advertisement to ensure maximal profit.
5) Write down the equation of the least squares line for the following
points
(2, 2)
(2, 3)
(5, 5).
a) (1, 2)
(0, 4)
(2, 3)
(4, 2)
b) (−2, 5)
(6, 1).
6) In the last 5 years a company sold every year a certain product according to the following table
year
sales
1
2
3
4 5
0.9 1.5 1.9 2.4 3
Use the least squares line to estimate how much will the company sale next
year.
2
Calculus B for Economics
Exercise Number 8
1) Use the mean value theorem to prove that if f 0 (x) = 0 for x, then
there exists a constant c such that f (x) = c.
2) Compute the following indefinite integrals
Z
Z
Z
dt
x+1
2
a) (x − 1) dx
b)
c)
dx
4
t
x
Z
Z 3
Z
x −1
x+2
d)
e dx
e)
dx
f ) (t − a)(t − b)dt
x2
Z
Z
√
√
g(x)f 0 (x) − g 0 (x)f (x)
dx
g) ( x + 1)(x − x + 1)dx h)
g 2 (x)
3) Compute the following definite integrals
Z10
a)
Za+2
b)
xdx
xdx
0
Z3
c)
a−2
Z0
x4
dx
3
d)
1
(a + t)2
dt
a
−a
Rx
t2 dt. Compute F (x) and F 0 (x).
R
R d
5) Is the identity
f (x)dx = dx
f (x)dx true for all functions f (x)?
R
R
In other words, is it true that ( f (x)dx)0 = f 0 (x)dx?
4) Define F (x) =
1
d
dx
6) Give an example to a nonzero continuous function f (x), and to an
Rb
interval [a, b], such that f (x)dx = 0.
a
1
7) Compute
Z1
Z−2
Z3
3
x dx + x dx + ex dx
2
−2
1
3
2
Calculus B for Economics
Exercise Number 9
1) In each of the following, sketch the region enclosed by the given curves.
Then compute the area of the region.
a) y = −x2 + 4, x = 0
√
c) y = x, y = x2
e) y = x3 − x,
b) y = x3 ,
y = x4 ,
d) y = x2 − x,
x≥0
y = −x2 + x
y = x4 − 1
2) Let f (x) be a continuous function on the interval [a, b]. Assume that
m ≤ f (x) ≤ M for all x ∈ [a, b]. Prove that
1
m≤
b−a
Zb
f (x)dx ≤ M
a
3) Let f (x) be a continuous function on the interval [a, b]. Assume that
Rb
f (x)dx = 0.
a
a) Is it necessarily true that f (x) = 0 for all x ∈ [a, b]?
b) Is it necessarily true that f (x) = 0 for some x ∈ [a, b]?
Rb
c) Is it necessarily true that | f (x)dx| = 0?
a
d) Is it necessarily true that
Rb
|f (x)|dx = 0.
a
Rb
e) Is it necessarily true that (f (x) + 1)dx = b − a?
4) Prove that
R2
1
a
dx
(x+1)20
≤ 1.
1
5) Which of the following two numbers is bigger:
Z1
Z1
x2
3
2x dx
2 dx
0
0
2
Calculus B for Economics
Exercise Number 10
1) Use a suitable substituting, or any other way, to compute the following
integrals
Z
1.1) (x + 1)15 dx
Z
x3 dx
√
1.4)
dx
3
x4 + 1
Z
x+3
1.7)
dx
(x − 4)2
Z
dx
1.10)
2x − 1
Z
(lnx)m
dx m 6= −1
1.13)
x
Z 2
x −1
1.16)
dx
x2 + 1
Z
1.2)
√
Z
8 − 2xdx
Z
1.5)
(3x + 1)
Z
1.8)
1.11)
1.14)
1.17)
√
1.3)
Z
3x2
+ 2x + 5dx 1.6)
Z
√
e x
√ dx
x
Z
dx
xlnx
Z
xdx
x+4
Z
dx
x2 + 3
√
5
x2 x3 + 2dx
1.9)
Z
(2x2
lnx2
dx
x
2
xex dx
1.12)
Z
3+x
dx
3−x
Z
x+4
1.18)
dx
x2 + 1
1.15)
2) Use integration by parts, or any other way, to compute the following
integrals
Z
Z
Z
ln3x
2x
−x
2.1)
xe dx
2.2)
xe dx
2.3)
dx
x
Z
Z
Z
2.4)
x2 lnxdx
2.5)
ln(x2 + 1)dx
2.6)
x2 ln(x + 1)dx
Z
2.7)
x2 e−x dx
3) Compute the area of the region enclosed by the following curves
a) y 2 = 2x − 1 and x − y − 1 = 0.
1
3x + 6
dx
+ 8x + 3)2
b) y = −x2 + 4x − 3 and the tangent lines to the parabola at the points
(0, −3) and (3, 0).
c) y 2 + 8x = 16 and y 2 − 24x = 4x.
4) Let n be an integer, and a a nonzero real number. Using integration
by parts, prove the identity
Z
Z
1 n ax n
n ax
x e dx = x e −
xn−1 eax dx
a
a
R
5) Use exercise 4) to compute the integral x3 e5x dx.
2
Calculus B for Economics
Exercise Number 11
1) In each of the following cases divide the polynomial f (x) by g(x)
1.1) f (x) = x3 + 3x2 − 1,
1.2) f (x) = x4 − 1,
g(x) = x + 1 1.2) f (x) = x4 ,
g(x) = x2 − 1
g(x) = x − 1
2) Using partial fractions, or any other way, compute the following integrals
Z
dx
2.1)
2
x −4
Z
x5 dx
2.4)
(x − 2)2
Z
dx
2.7)
2
x + 16
Z
xdx
2.10)
x4 + 1
Z
xdx
2.2)
(x + 1)(x + 2)(x + 3)
Z
(x2 + 3)dx
2.5)
x2 − 3x + 2
Z
x2 dx
2.8)
(x − 1)2 (x + 1)
Z
(2x2 + 41x − 91)dx
2.11)
(x − 1)(x + 3)(x − 4)
3) Compute the following integrals. Enjoy!
Z
Z
√
ln3x
3.1)
dx
3.2)
x
2x + 1dx
x2
Z −√x
Z
e
xdx
√ dx
3.4)
3.5)
(x + 1)2
x
Z
Z
√
xln(ax + b)dx a 6= 0
3.7)
ln(x x)dx 3.8)
Z
Z
dx
ex dx
√
√
3.11)
3.10)
ex + 1
1 − e2x
1
Z
x5 dx
x−2
Z
dx
2.6)
(x − 1)3
Z
x3 dx
2.9)
x4 + 1
2.3)
Z
dx
√
x+1− x
Z
√
3.6)
xln x2 + 1dx
Z
(x − 3)dx
3.9)
x2 (x + 1)
Z √
3.12)
e x dx
3.3)
√
Calculus B for Economics
Exercise Number 12
3
1) Verify that y = e−x + ce− 2 x solves the equation 2y 0 + 3y = e−x . Find
the value of c if y(0) = 2.
2) Solve the following differential equations. If an initial value is given,
find the constant c.
2.2) xy 2 dx + (x2 + 1)dy = 0
√
√
2.4) xy 0 = ey+ x
1
2.6) y 0 = y(1 − y) y(0) =
3
2.1) (y + 2)dx + (x − 2)dy = 0
2.3) (1 − x)y 0 = y 2
e2x−y
2.5) y 0 = x+y
e
3) The price of a certain product is 3 Shekels per kg. It is estimated that
√
x days from today the price will increase at the rate of 3 x + 1 Agorot per
day. What will be the price of the product in eight days from today?
4) Solve the following linear differential equations
dy y
+ = 3x
dx x
4.3) xy 0 − ny = xn+1 n 6= 0
4.2) − y = e2x − y 0
4.1)
4.4) xy 0 − y = 2xlnx
5) A tank contains 2000 liters of gasoline which includes 100 kg of an
additive in it. Every minute 40 liters of mixture which contains 2 kg of
additive per liter, is inserted into the tank, and every minute 45 liters of
mixture is drained out. How much additive is in the tank after 20 minutes?
1
Calculus B for Economics
Exercise Number 13
1) Verify that the following equations are homogenous and solve them.
1.1) (x2 + y 2 )dx − xydy = 0
1.2) (3x + y) − (x + 3y)y 0 = 0 y(4) = 2
2) Verify that the following equations are exact and solve them.
2.1) (3x2 y 2 + 2y 3 )dx + (2x3 y + 6xy 2 )dy = 0 2.2) (2x + 3y) + (3x − 4y)y 0 = 0
3) Solve the following differential equations.
3.1) (1 + y 2 )dx + xdy = 0
3.2) (x + y + 1)dx + (x − y − 1)dy
4) Consider the equation ydx + dy = 0. Show that it is not an exact
equation. Multiply by ex and obtain yex dx + ex dy = 0. Show that this
equation is exact and solve it. The function ex is called an integrating factor.
5) Consider the differential equation M (x, y)dx + N (x, y)dy = 0. If we
multiply it by a given function µ(x) > 0 we obtain the equation µ(x)M (x, y)dx+
µ(x)N (x, y)dy = 0. Show that this equation is exact if and only if µ(x) satisfies the equation
∂M
∂N
= µ0 (x)N + µ(x)
∂y
∂x
Show that this is equivalent to the equation
µ(x)
∂M
∂y
−
∂N
∂x
N
=
µ0 (x)
µ(x)
Notice that the right hand side is a function of x only. Hence deduce the
following:
1
The equation µ(x)M (x, y)dx + µ(x)N (x, y)dy = 0 is exact if and only if
the function
∂M
∂y
− ∂N
∂x
N
∂M
∂y
− ∂N
∂x
N
is a function of x only. Assuming that, denote f (x) =
R
. Prove that µ(x) = e
f (x)dx
. Such a µ(x) is called an integrating
factor.
6) Use exercise 5) to solve the following equations.
6.1) (3xy + y 2 ) + (x2 + xy)y 0 = 0
2
6.2) y 0 = e2x + y − 1