Q16Mathematics II

Question

Q.6 Solve : $9 (p^{2} z + q^{2}) = 4$

Answer 1

Given the explicitly non-linear partial differential equation:

9 (p^{2} z + q^{2}) = 4

Step 1: Standard Substitution

Since this exact equation completely lacks independent variables $x$ and $y$ explicitly, it strictly falls under the standard form $f (z, p, q) = 0$ . The absolutely standard rigorous method is to assume a solution of the form $z = f (u)$ where $u = x + a y$ .

This rigorously implies:

p = \frac{\partial z}{\partial x} = \frac{d z}{d u} \frac{\partial u}{\partial x} = \frac{d z}{d u} (1) = \frac{d z}{d u}

q = \frac{\partial z}{\partial y} = \frac{d z}{d u} \frac{\partial u}{\partial y} = \frac{d z}{d u} (a) = a \frac{d z}{d u}

Step 2: Conversion to ODE

We systematically substitute these exact expressions directly back into the given PDE:

9 [z (\frac{d z}{d u})^{2} + (a \frac{d z}{d u})^{2}] = 4

9 (\frac{d z}{d u})^{2} (z + a^{2}) = 4 ⟹ (\frac{d z}{d u})^{2} = \frac{4}{9 ( z + a ^{2} )}

Step 3: Solving the ODE

Taking the exact positive square root of both sides:

\frac{d z}{d u} = \frac{2}{3 z + a ^{2}}

We rigidly separate variables completely:

z + a^{2} d z = \frac{2}{3} d u

Integrating exactly both sides mathematically:

\int (z + a^{2})^{1/2} d z = \int \frac{2}{3} d u

\frac{( z + a ^{2} ) ^{3/2}}{3/2} = \frac{2}{3} u + C

\frac{2}{3} (z + a^{2})^{3/2} = \frac{2}{3} (x + a y) + C

Step 4: Final Simplification

Multiplying entirely by $3/2$ and replacing $3 C /2$ strictly with a new arbitrary constant $c$ :

(z + a^{2})^{3/2} = (x + a y) + c

Squaring both sides explicitly to remove the fractional power mathematically yields the absolute final solution:

(z + a^{2})^{3} = (x + a y + c)^{2}

Q16Mathematics II

Question

Answer

Step 1: Standard Substitution

Step 2: Conversion to ODE

Step 3: Solving the ODE

Step 4: Final Simplification