Answer to Question #296560 in Differential Equations for Ruhi

Question

Solve:

(𝑥 − 2𝑠𝑖𝑛𝑦 + 3)𝑑𝑥 + (2𝑥 − 4𝑠𝑖𝑛𝑦 − 3)𝑐𝑜𝑠𝑦𝑑𝑦 = 0

Answer 1

$\begin{gathered} (x-2 \sin y+3) d x+(2 x-4 \sin y-3) \cos y d y=0 \\ u=\sin y \\ d u=\cos y d y \Rightarrow d y=d u / \cos y \end{gathered}$

Substituting in differential equation.

 $\begin{aligned} &(x-2 u+3) d x+(2 x-4 u-3) \cos y \frac{d u}{\cos y}=0 \\ & \text { Integrating } \\ \Rightarrow & \int(x-2 u+3) d x+\int(2 x-4 u-3) d u=C \\ \Rightarrow & \frac{x^{2}}{2}-2u x+3 x+\frac{2 x^{2}}{2}-4u x-3 u=C \\ \Rightarrow & \frac{3 x^{2}-64 x-3 u+3 x=C}{2}\\\Rightarrow & \frac{3 x^{2}}{3}-6 x \sin y-3 \sin y+3 x=C \end{aligned}$