Variation of Parameters and
Euler Equation
Non- homogeneous linear d.e with a)constant
coefficients and b) variable coefficients
METHOD OF VARIATION OF PARAMETERS (MVP)
 This method is used when the method of undetermined coefficients
will not work.
Applied to any linear equation of any order with constant or variable
coefficients regardless of the forms of f(x) and for which yc is
available.
 Consider the 2nd order linear differential equation:
𝑦 ′′ + 𝑃 𝑥 𝑦 ′ + 𝑄 𝑥 𝑦 = 𝑓(𝑥)
Solution: y = yc + yp
yc= c1y1 + c2y2
yp = u1y1 + u2y2
Assume u1 and u2 as functions of x
METHOD OF VARIATION OF PARAMETERS (MVP)
PROCEDURE
 Solve for yc
Replace c’s in yc with u’s to get yp
Set up the Wronskian matrix
Solve the u1’, u2’,……un’ (via Cramer’s Rule)
Obtain u1, u2…..un by integration to get final yp
CAUCHY- EULER LDE
Standard form:
𝑛
𝑑
𝑦
𝑛
𝑎𝑛 𝑥
+
𝑛
𝑑𝑥
𝑛−1
𝑑
𝑦
𝑑𝑦
𝑛−1
𝑎𝑛−1 𝑥
+ … . . + 𝑎1 𝑥
+ 𝑎0 𝑦 = 𝑓(𝑥)
𝑛−1
𝑑𝑥
𝑑𝑥
Where a0 , a1 , ….. an
𝑑
𝐷=
𝑑𝑥
Example:
are constants
𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟
𝑥 2 𝐷2 − 𝑥𝐷 + 1 𝑦 =
1
𝑥
NOTES: 1) Special LDE with variable coefficients
2) Powers of x and D are equal in each term
CAUCHY- EULER LDE
• Transformation
Let z = ln x
→
𝒙 = 𝒆𝒛
Notes: 𝐷𝑥 =
𝑑
𝑑𝑥
dz/dx = 1/x
• For the first derivative:
𝐷𝑥 𝑦
𝐷𝑥 𝑦
𝑑𝑦
𝑑𝑦 𝑑𝑧
=
=
.
𝑑𝑥
𝑑𝑥 𝑑𝑧
𝑑𝑧 𝑑𝑦
=
.
𝑑𝑥 𝑑𝑧
1 𝑑𝑦
1
=
.
=
𝐷𝑧 𝑦
𝑥 𝑑𝑧
𝑥
x D x y = 𝑫𝒛 𝒚
𝐷𝑦 =
𝑑
𝑑𝑦
CAUCHY- EULER LDE
Transformation
For the second derivative
𝑑
𝐷𝑥2 𝑦=
𝐷𝑥 𝑦 =
𝑑𝑥
𝑑 1
𝑑𝑥 𝑥
=
=
=
=
•
1 𝑑
𝑥 𝑑𝑥
1 𝑑
𝑥 𝑑𝑥
𝐷𝑧 𝑦 =
1
𝐷𝑦
𝑥2 𝑧
𝑑𝑧
1
− 2 𝐷𝑧
𝑑𝑧
𝑥
𝐷𝑧 𝑦 −
𝐷𝑧 𝑦
1 𝑑
𝑑𝑧
1
𝐷𝑧 𝑦
− 2 𝐷𝑧 𝑦
𝑥 𝑑𝑧
𝑑𝑥
𝑥
1 2
1
1
𝐷 𝑦
− 2 𝐷𝑧 𝑦
𝑥 𝑧
𝑥
𝑥
1
2𝑦 − 𝐷 𝑦
𝐷
𝑧
𝑧
𝑥2
𝑥 2 𝐷𝑥2 𝑦 = 𝐷𝑧2 𝑦 − 𝐷𝑧 𝑦
𝒙𝟐 𝑫𝟐𝒙 𝒚 = 𝑫𝒛 𝑫𝒛 − 𝟏 𝒚
CAUCHY- EULER LDE
• Transformatio n
For the third derivative:
• x Dx y = 𝐷𝑧 𝑦
• 𝑥 2 𝐷𝑥2 𝑦 =
𝐷𝑧 𝐷𝑧 − 1 𝑦
• 𝒙𝟑 𝑫𝟑𝒙 𝒚 =
𝑫𝒛 𝑫𝒛 − 𝟏 (𝑫𝒛 − 𝟐 𝒚
• 𝑥 𝑛 𝐷𝑥𝑛 𝑦 =
𝐷𝑧 𝐷𝑧 − 1 𝐷𝑧 − 2 … … . 𝐷𝑧 − 𝑛 − 1
𝑦
CAUCHY- EULER LDE
PROCEDURE:
 Make sure that the given LDE with variable coefficients conforms to
the Cauchy-Euler form.
Substitute the boxed expressions in the given differential equation
Solve a transformed LDE (now with constant coefficients) using MUC
or MVP with z as the independent variable
Resubstitute x= ez or z = ln 𝑥 𝑡𝑜 𝑜𝑏𝑡𝑎𝑖𝑛 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑥