Q8Mathematics II

Question

Q.8 Solve : $a (p + q) = z$

Answer 1

The given partial differential equation is exactly $a p + a q = z$ , which strictly conforms to Lagrange's standard linear form $P p + Qq = R$ , where $P = a$ , $Q = a$ , and $R = z$ .

The corresponding Lagrange's auxiliary differential equations are rigidly formulated as:

\frac{d x}{P} = \frac{d y}{Q} = \frac{d z}{R} ⟹ \frac{d x}{a} = \frac{d y}{a} = \frac{d z}{z}

First Integral

Taking the exact first two mathematical fractions:

\frac{d x}{a} = \frac{d y}{a} ⟹ d x = d y ⟹ \int d x = \int d y

x = y + c_{1} ⟹ u = x - y = c_{1}

Second Integral

Taking the strictly first and exact last mathematical fractions:

\frac{d x}{a} = \frac{d z}{z} ⟹ \int \frac{d x}{a} = \int \frac{d z}{z}

\frac{x}{a} = ln (z) - ln (c_{2}) ⟹ ln (\frac{z}{c _{2}}) = \frac{x}{a}

\frac{z}{c _{2}} = e^{x / a} ⟹ v = z e^{- x / a} = c_{2}

General Solution

The absolute general solution is mathematically expressed as an arbitrary function $f (u, v) = 0$ :

f (x - y, z e^{- x / a}) = 0

Q8Mathematics II

Question

Answer

First Integral

Second Integral

General Solution