Cheat Sheet – Differential Equations
Variable Separable
Reducible to Variable Separable
Form
Form
dy
= F(x). G(y)
dx
dy
= f(ax + by + c)
dx
Solution
Solution
dy
Separate the variables:
= F(x)dx
G(y)
Substitute ax + by + c = t ⇒ a + b
Integrate both sides: ∫
dy dt
=
dx dx
1 dt
( − a) = f(t)
b dx
dt
Separate the variables:
= dx
bf(t) + a
dy
= ∫ F(x)dx
G(y)
Convert the equation:
Integrate both sides: ∫
dt
= ∫ dx
bf(t) + a
Homogeneous
Reducible to Homogeneous
Form
Form
dy
y
= f( )
dx
x
dy ax + by + c
=
;
dx px + qy + r
Solution
Solution
dv dy
Put y = vx ⇒ v + x =
dx dx
Put x = X + h, y = Y + k
dv
= f(v)
dx
dv
dx
Separate the variables:
=
f(v) − v
x
where ah + bk + c = 0, ph + qk + r = 0
Convert the equation: v + x
Integrate both sides: ∫
Convert the equation:
dv
dx
=∫
f(v) − v
x
a b
≠
p q
dY aX + bY
=
dX pX + qY
Solve as a homogeneous differential equation
Linear
Reducible to Linear
Form
Form
dy
+ P(x)y = Q(x)
dx
f′(y)
dy
+ P(x)f(y) = Q(x)
dx
Solution
Solution
Calculate Integrating Factor: I(x) = e∫ P(x)dx
Substitute f(y) = t ⇒ f ′ (y)
Solution: y. I(x) = ∫ Q(x). I(x)dx
Convert the equation:
dy dt
=
dx dx
dt
+ P(x)t = Q(x)
dx
Solve as a linear differential equation
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