First slide

Methods of solving first order first degree differential equations

Question

$If x d y = (2 y + 2 x^{4} + x^{2}) d x, y (1) = 0 then y (e) =$

Moderate

A

1
B

e
C

e²
D

e⁴

Solution

$\begin{array}{l} \frac{d y}{d x} - \frac{2}{x} y = 2 x^{3} + x \\ I.F = \exp \int \frac{- 2}{x} d x = \frac{1}{x^{2}} \\ Sol is \frac{y}{x^{2}} = \int (2 x + \frac{1}{x}) d x = x^{2} + \ln x + c \\ x = 1, y = 0 \to c = - 1 \\ y (x) = x^{4} + x^{2} \ln (x) - x^{2} \\ y (e) = e^{4} \end{array}$

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