Question

Solve (p – x) (p – e^x) (p – 1/y) = 0 ; where p = $\frac{dy}{dx}$ :

• A. (y – $\frac{x^{2}}{2}$ + c) (y – e^x + c) ($\frac{y^{2}}{2}$ – x + c) = 0
• B. (y + $\frac{x^{2}}{2}$ + c) (y – e^x + c) ($\frac{y^{2}}{2}$ – x + c) = 0
• C. (y – $\frac{x^{2}}{2}$ + c) (y – e^x + c) ($\frac{y^{2}}{2}$ + x + c) = 0
• D. None of these

Answer 1

Answer: (A)

(y – $\frac{x^{2}}{2}$ + c) (y – e^x + c) ($\frac{y^{2}}{2}$ – x + c) = 0

Answer 2

The component linear equations are ;

p = x, p = e^x , p = 1/y

If $\frac{dy}{dx}$ = x then dy = x dx Ž y = $\frac{x^{2}}{2}$ + c₁

If $\frac{dy}{dx}$ = e^x then dy = e^x dx Ž y = e^x + c₂

If $\frac{dy}{dx}$ = $\frac{1}{y}$ then y dy = dx Ž $\frac{y^{2}}{2}$ = x + c₃

\ The required solution is ;

$\left( y - \frac{x^{2}}{2} + c \right)$ (y – e^x + c) $\left( \frac{y^{2}}{2} - x + c \right)$ = 0