Q12Mathematics II

Question

Q.2 Solve : $y = 2 p x + p^{2} y$

Answer 1

Given the differential equation explicitly containing $p$ ( $p = \frac{d y}{d x}$ ):

y = 2 p x + p^{2} y

Step 1: Rearrangement for Differentiation

Since $x$ appears purely as a single linear term, we mathematically solve the equation exactly for $x$ :

2 p x = y - p^{2} y = y (1 - p^{2})

x = \frac{y ( 1 - p ^{2} )}{2 p} = \frac{y}{2} (\frac{1}{p} - p)

Step 2: Differentiating with respect to $y$

We strictly differentiate the entire equation completely with respect to $y$ . We meticulously recall that $\frac{d x}{d y} = \frac{1}{p}$ .

\frac{d x}{d y} = \frac{1}{2} [1 \cdot (\frac{1}{p} - p) + y \cdot \frac{d}{d y} (\frac{1}{p} - p)]

\frac{1}{p} = \frac{1}{2} [\frac{1}{p} - p + y (- \frac{1}{p ^{2}} \frac{d p}{d y} - \frac{d p}{d y})]

\frac{2}{p} = \frac{1}{p} - p - y (\frac{1}{p ^{2}} + 1) \frac{d p}{d y}

\frac{1}{p} + p + y (\frac{1 + p ^{2}}{p ^{2}}) \frac{d p}{d y} = 0

Mathematically factoring out strictly $(\frac{1 + p ^{2}}{p})$ :

(\frac{1 + p ^{2}}{p}) (1 + \frac{y}{p} \frac{d p}{d y}) = 0

Step 3: Solving the Separable Equation

We completely ignore the factor $\frac{1 + p ^{2}}{p}$ as it strictly leads exactly to the singular solution. We rigidly equate the second differential factor exactly to zero:

1 + \frac{y}{p} \frac{d p}{d y} = 0

This is a simple separable first-order ODE. We strictly separate variables:

\frac{d p}{p} + \frac{d y}{y} = 0

Integrating both sides exactly yields:

ln p + ln y = ln c ⟹ ln (p y) = ln c ⟹ p y = c ⟹ p = \frac{c}{y}

Step 4: Final Substitution

To explicitly find the general solution, we rigorously substitute this exact value of $p$ completely back into the original given equation:

y = 2 (\frac{c}{y}) x + (\frac{c}{y})^{2} y

y = \frac{2 c x}{y} + \frac{c ^{2}}{y}

Multiplying the entire equation exactly by $y$ yields the absolute final general solution:

y^{2} = 2 c x + c^{2}

Q12Mathematics II

Question

Answer

Step 1: Rearrangement for Differentiation

Step 2: Differentiating with respect to y

Step 3: Solving the Separable Equation

Step 4: Final Substitution

Step 2: Differentiating with respect to $y$