The document discusses methods for solving partial differential equations (PDEs) of the form pP + qQ = R, where P, Q, R are functions of x, y, and z. It describes the subsidiary equation and two solution methods: the method of grouping and the method of multipliers. The method of multipliers involves choosing Lagrangian multipliers l, m, n such that lP + mQ + nR = 0, which yields an exact differential equation that can be integrated to find the solution.

1. The pde is in the form 𝒑𝑷 + 𝒒𝑸 = 𝑹,
where 𝑷, 𝑸, 𝑹 are the functions of
𝒙, 𝒚 & 𝒛

2. Note
1. 𝑃 =
𝑢 𝑦 𝑢 𝑧
𝑣 𝑦 𝑣𝑧
= uyvz − uzvy
2. 𝑄 =
𝑢 𝑥 𝑢 𝑧
𝑣 𝑥 𝑣𝑧
= u 𝑥vz − uzv 𝑥
3. 𝑅 =
𝑢 𝑥 𝑢 𝑦
𝑣 𝑥 𝑣 𝑦
= u 𝑥vy − uyv 𝑥
4.The subsidiary (or) auxiliary equation of given
pde is
𝑑𝑥
𝑃
=
𝑑𝑦
𝑄
=
𝑑𝑧
𝑅

3. There are two methods to solve the given pde
(i) Method of grouping
(ii) Method of multipliers

5. Method of multipliers
 First write the auxiliary (or) subsidiary equation of the given pde
𝑑𝑥
𝑃
=
𝑑𝑦
𝑄
=
𝑑𝑧
𝑅
 Choose the multipliers 𝑙, 𝑚, 𝑛 such that
𝑑𝑥
𝑃
=
𝑑𝑦
𝑄
=
𝑑𝑧
𝑅
=
𝑙𝑑𝑥+𝑚𝑑𝑦+𝑛𝑑𝑧
𝑙𝑃+𝑚𝑄+𝑛𝑅
 The multipliers 𝑙, 𝑚, 𝑛 are called Lagrangian multipliers.
 If it is possible to choose 𝑙, 𝑚, 𝑛 such that 𝑙𝑃 + 𝑚𝑄 + 𝑛𝑅 = 0
then 𝑙𝑑𝑥 + 𝑚𝑑𝑦 + 𝑛𝑑𝑧 = 0.
 The equation 𝑙𝑑𝑥 + 𝑚𝑑𝑦 + 𝑛𝑑𝑧 = 0 is an exact differential
equation; on integration we will get the solution. 𝑢(𝑥, 𝑦) =
𝑎 & 𝑣(𝑥, 𝑦) = 𝑏
 Final solution is ∅ 𝑢, 𝑣 = 0, where ∅ is arbitrary