First slide

Types of solutions of differential equations

Question

$The solution of d.E (x^{2} + y^{2} + x) d x - (2 x^{2} + 2 y^{2} - y) d y = 0 is$

Moderate

A

$x - 2 y + \ln (x^{2} + y^{2}) = c$
B

$x - 2 y + \ln (x^{2} - y^{2}) = c$
C

$x - 2 y + \ln (x^{2} + 2 y^{2}) = c$
D

$x - 2 y + \frac{1}{2} \ln (x^{2} + y^{2}) = c$

Solution

$(x^{2} + y^{2}) d x - 2 (x^{2} + y^{2}) d y + (x d x + y d y) = 0$
$\begin{array}{l} \Rightarrow d x - 2 d y + \frac{x d x + y d y}{x^{2} + y^{2}} = 0 \\ Integrations both sides x - 2 y + \frac{1}{2} \ln (x^{2} + y^{2}) = c \end{array}$

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