First slide

Introduction to Differential equations

Question

$If \frac{d y}{d x} = x y + 2 x + 3 y + 6 then y (- 1) - e^{2} y (- 3) =$

Moderate

A

$e^{2} - 1$
B

$e^{2} + 1$
C

$2 (e^{2} - 1)$
D

$- 2 (e^{2} - 1)$

Solution

$\begin{array}{l} \frac{d y}{d x} = (x + 3) (y + 2) \\ X = x + 3, Y = y + 2 \Rightarrow \frac{d t}{Y} = X \cdot d X \\ y = e^{A \frac{x^{2}}{2}}, y = - 2 + A \cdot e^{(x + 3)^{2}} / 2 \\ y (- 1) = - 2 + A e^{2} \\ y (- 3) = - 2 + A \end{array}$
$y (- 1) - e^{2} x (- 3) = 2 (e^{2} - 1)$

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