432
CHAPTER 46
46.10
Solve
46.11
Solve
46.12
Solve
where C = tan C,.
tan""
1 y = tan~' x + C,,
e
3y = 3ln\l + x\ + C,
3y = In(3In |l + *| + C),
y =
46.13
Solve
y = sin (sin ' x + C) = sin (sin * AC) cos C + cos (sin * x) sin C = x cos C + Vl — x
2 sin C
46.14
Solve
given
Since
C = -2.
Hence,
46.15
Solve
46.16
Solve x dx + y dy = xX* dy — y dx).
46.17 Solve / = (* + y)
2
.
[Tie variables do not separate. Try the substitution z = x + y. Then, z' = l + y'. So z' -1 = z
2
,
Now, the variables x and z are separable.
* + >> = tan (x + C), y = tan (A: + C) - x.
tan ' z = x + C, z = tan (x + C),
46.18
Solve
The variables are not separable. However, on the right side, the numerator and denominator are
homogeneous of the same degree (one) in x and y. In such a case, let y = vx. Then,
Hence,
Thus,
Now the variables x and
sin ' y = sin ' x + C,
As usual