Unit V: Simultaneous and Total Differential Equations

Simultaneous Differential Equations

Simultaneous differential equations consist of two or more differential equations containing more than one dependent variable but only a single independent variable. The objective is to determine the dependent variables as explicit functions of the independent variable.

Types of Simultaneous Systems

We generally encounter two main types of simultaneous differential equations in undergraduate courses:

1. Linear Simultaneous Differential Equations: Systems involving derivative operators, typically expressed with respect to an independent variable (often t). For example:

   dx/dt + 2x - 3y = e^t
   dy/dt - 3x + 2y = 0

2. Symmetrical Simultaneous Differential Equations: Systems of the form:

   dx / P = dy / Q = dz / R

   where P, Q, and R are functions of x, y, and z.

Symmetrical Form and Methods of Solution

To solve symmetrical systems of the form dx/P = dy/Q = dz/R, we use two primary techniques:

1. Method of Grouping (Combinations)

In this method, we select any two ratios such that the third variable is either completely absent or can be algebraically eliminated. This allows us to form an ordinary differential equation with only two variables, which can be solved directly by separation of variables or other standard first-order methods.

Step-by-Step Procedure:

1. Select a pair of ratios, say dx/P = dy/Q.
2. If the equation contains only x and y (or if z cancels out), integrate it to get a relation:

   u(x, y) = c₁

3. Select another pair, such as dy/Q = dz/R or dx/P = dz/R. If it contains only the two relevant variables, integrate it to get:

   v(y, z) = c₂ (or v(x, z) = c₂)

4. The two solutions together, u(x, y) = c₁ and v(y, z) = c₂, constitute the complete general solution of the system.

2. Method of Multipliers

When grouping is not directly possible because variables cannot be separated or eliminated, we find a set of multipliers l, m, n (which may be constants or functions of x, y, z) such that:

(l dx + m dy + n dz) / (lP + mQ + nR) = dx/P = dy/Q = dz/R

If we can find multipliers l, m, n such that the denominator becomes zero:

lP + mQ + nR = 0

Then the numerator must also be zero to keep the ratio finite:

l dx + m dy + n dz = 0

If the expression l dx + m dy + n dz is an exact differential (or can be integrated easily after multiplying by an integrating factor), we can integrate it to find one part of the solution:

u(x, y, z) = c₁

By finding a second set of multipliers l', m', n' such that l'P + m'Q + n'R = 0, we can obtain another independent solution:

v(x, y, z) = c₂

Total Differential Equations (Pfaffian Form)

A total differential equation (or Pfaffian differential equation) in three variables is of the form:

P dx + Q dy + R dz = 0

Condition for Integrability

A total differential equation P dx + Q dy + R dz = 0 is not always solvable. It is solvable if and only if there exists a family of surfaces u(x, y, z) = c whose tangent planes satisfy the equation at every point. This property is known as integrability.

The Mathematical Condition

The necessary and sufficient condition for the integrability of the equation P dx + Q dy + R dz = 0 is:

P(∂Q/∂z - ∂R/∂y) + Q(∂R/∂x - ∂P/∂z) + R(∂P/∂y - ∂Q/∂x) = 0

Vector Form Interpretation

If we define a vector field F = P i + Q j + R k, then the condition of integrability is the vanishing of the scalar triple product of F and its curl:

F · (curl F) = 0

Condition for Exactness

A total differential equation P dx + Q dy + R dz = 0 is said to be exact if its left-hand side is the exact total differential of some function u(x, y, z). That is:

du = P dx + Q dy + R dz = 0

The Conditions for Exactness

The necessary and sufficient conditions for the equation to be exact are:

∂P/∂y = ∂Q/∂x
∂Q/∂z = ∂R/∂y
∂R/∂x = ∂P/∂z

Difference between Integrability and Exactness

Methods of Solving Total Differential Equations

Once the condition of integrability is verified and satisfied, we can solve the total differential equation using several methods depending on the structure of P, Q, and R.

Method 1: Integration by Inspection

If the terms of the equation can be grouped and rearranged to form exact differentials of known algebraic functions, we can integrate directly.

Common Differential Groupings:

• x dy + y dx = d(xy)
• (y dx - x dy) / y² = d(x/y)
• (x dx + y dy + z dz) = (1/2) d(x² + y² + z²)
• (x dy - y dx) / (x² + y²) = d(arctan(y/x))

Method 2: One Variable Treated as a Constant (Auxiliary Variable Method)

This is a highly reliable systematic method when the equation is integrable but not easily groupable by inspection.

Step-by-Step Procedure:

1. Assume one of the variables, say z, is constant. Therefore, its differential is zero: dz = 0.
2. Substitute dz = 0 into the original equation, reducing it to:

   P dx + Q dy = 0

3. Solve this two-variable differential equation. The constant of integration, instead of being a simple constant, must be treated as a function of the held-constant variable z. Let this solution be:

   u(x, y) = φ(z)

4. Take the total differential of this solution:

   (∂u/∂x) dx + (∂u/∂y) dy = dφ

5. Compare this result with the original equation to determine the expression for dφ/dz.
6. Integrate dφ/dz with respect to z to obtain φ(z), and substitute it back into the equation to get the final complete solution.

Method 3: Homogeneous Equations Method

If P(x,y,z), Q(x,y,z), and R(x,y,z) are homogeneous functions of the same degree n, the equation can be solved using substitution.

Steps:

1. Verify homogeneity of the coefficients.
2. Substitute:

   x = uz and y = vz

   Which yields:

   dx = u dz + z du and dy = v dz + z dv

3. Substitute these into the main equation and group the terms of du, dv, and dz. The variable z will separate out, allowing integration.
4. Integrating Factor Rule: Alternatively, if the equation is homogeneous and xP + yQ + zR ≠ 0, then:

   Integrating Factor (I.F.) = 1 / (xP + yQ + zR)

   Multiplying the total differential equation by this factor transforms it into an exact differential equation.

Common Student Errors & Exam Tips

• Mistake: Jumping straight to solving without testing for integrability. Always check the integrability condition first in examinations unless the question explicitly states that the equation is integrable.
• Mistake: Incorrect partial differentiation when evaluating the integrability condition. Remember that when differentiating with respect to x, both y and z are treated as constants.
• Tip: If the coefficients P, Q, R are symmetric, look closely for multipliers like 1, 1, 1 or x, y, z or 1/x, 1/y, 1/z. These are highly common in academic exams.