Lagrange’s Linear Equation PDF

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1/29/22, 10:37 PM

Lagrange’s Linear Equation

Chapter: Mathematics (maths) - Partial Differential Equations

Lagrange’s Linear Equation

Lagrange’s Linear Equation Equations of the form Pp + Qq = R

(1), where P, Q and R are functions of x, y, z, are known as Lagrang solve this

equation, let us consider the equations u = a and v = b, where a, b are arbitrary constants and u, v are functions of x, y, z.

1/6

Equations (5) represent a pair of simultaneous equations which are of the first order and of first degree.Therefore, the two solutions of (5) are u = a and v = b. Thus, ( u, v ) = 0 is the required solution of (1). Note : To solve the Lagrange‟s equation,we have to form the subsidiary or auxiliary equations

which can be solved either by the method of grouping or by the method of multipliers. Example 21 Find the general solution of px + qy = z.

Here, the subsidiary equations are

Integrating, log x = log y + log c1 or x = c1 y i.e, c 1 = x / y From the last two ratios,

Integrating, log y = log z + log c2 or y = c2 z i.e, c2 = y / z Hence the required general solution is Φ( x/y,= 0,y/z)where Φ is arbitrary

Example 22 Solve p tan x + q tan y = tan z The subsidiary equations are

Hence the required general solution is

where Φ is arbitrary

Example 23 Solve (y-z) p + (z-x) q = x-y Here the subsidiary equations

are

Example 24 Find the general solution of (mz - ny) p + (nx- lz)q = ly - mx.

Exercises Solve the following equations 1.

px2 + qy2 = z2

2. pyz + qzx = xy 3. xp –yq = y2 –x2 4. y2 zp + x 2zq = y 2 x 5. z (x –y) = px2 –qy 2 6. (a –x) p + (b –y) q = c –z 7. (y 2z p) /x + xzq = y 2 8. (y2 + z 2 ) p –xyq + xz = 0 9. x2 p + y 2q = (x + y) z 10.

p –q = log (x+y)

11.

(xz + yz)p + (xz –yz)q = x2 +

y2 12.

(y –z)p –(2x + y)q = 2x + z...