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2x dy/dx = 2y - y² , where y = zx

y = zx

Differentiate both sides with respect to x,

dy/dx = (dz/dx)x + z(1) = x dz/dx + z

Sub this and y = zx into the original equation,

2x (x dz/dx + z) = 2(zx) - (zx)²

2x² dz/dx + 2xz = 2zx - z²x²

2x² dz/dx = -z²x²

2 dz/dx = -z²

-2/z² dz/dx = 1

Integrate both sides with respect to x,

∫ (-2/z² dz/dx) dx = ∫ 1 dx

∫ -2/z² dz = ∫ 1 dx

2/z = x + c

z = 2/(x + c)

Sub z = y/x, (because y = xz → z = y/x)

y/x = 2/(x + c)

y = 2x/(x + c)

(Solved)

dy/dx + y = y², where z = 1/y and y = ½ when x = 0

z = 1/y → y = 1/z

Differentiate both sides with respect to x,

(1) dy/dx = (-1/z²) dz/dx

dy/dx = -1/z² dz/dx

Sub this and y = 1/z into the original equation,

-1/z² dz/dx + 1/z = (1/z)²

1/z² dz/dx = 1/z - 1/z²

Multiply both sides by z²,

dz/dx = z - 1

1/(z - 1) dz/dx = 1

Integrate both sides with respect to x,

∫ (1/(z - 1) dz/dx) dx = ∫ 1 dx

∫ 1/(z - 1) dz = ∫ 1 dx

ln|z - 1| = x + c

|z - 1| = eˣ+ᶜ

|1/y - 1| = eˣ+ᶜ


Sub x = 0, y = ½,

|1/½ - 1| = eᶜ

ln|1| = eᶜ

eᶜ = 0


So,

|1/y - 1| = eˣ

1/y - 1 = eˣ or 1/y - 1 = -eˣ

1/y = eˣ + 1 or 1/y = 1 - eˣ (rejected as when x = 0, 1 - eˣ = 1 - 1 = 0. But when y = ½, 1/y = 2. This results in a contradiction)

y = 1/(eˣ + 1)

(Solved)

dy/dx = - (8x + 4y + 1)/(4x + 2y + 1) and y = u - 2x


y = u - 2x →u = 2x + y

Differentiate both sides with respect to x,

du/dx = 2 + (1) dy/dx
du/dx = 2 + dy/dx
dy/dx = du/dx - 2

Sub this and u = 2x + y into the original equation,

du/dx - 2 = -(4u + 1)/(2u + 1)

(2u + 1) du/dx - 2(2u + 1) = -(4u + 1)

(2u + 1) du/dx - 4u - 2 = -4u - 1

(2u + 1) du/dx = 1


Integrate both sides with respect to x,


∫ (2u + 1) du/dx = ∫ 1 dx

∫ (2u + 1) du = ∫ 1 dx

u² + u = x + c, c is a constant

Sub u = 2x + y,

(2x + y)² + 2x + y = x + c

(2x + y)² + x + y = c


The answer uses A as the letter for the constant but you can use other letters too like b,c,d,and their upper case forms.


Try not to use the following to avoid confusion or uncertainty :


x,y : They will be confused with the terms in the question itself.

w,z : usually reserved for complex numbers

u : used in substitutions, recurrence relations

i : imaginary unit

e : mathematical constant

f,g,h : used to denote functions

j : used in vectors (both 3D and 2D)

k : vectors as well as constants for trigo and complex numbers(roots of unity part)
but k is still acceptable

l : used for lines in vectors, shortform for litres

m : metres, milli-

o : origin, centre of circle, often confused with 0 as well.

r : radius of circle , summation , APGP

s : second

t : time, differentiation with respect to time,

v : velocity