Question #296560

Solve:


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

1
Expert's answer
2022-02-16T17:09:57-0500

(x2siny+3)dx+(2x4siny3)cosydy=0u=sinydu=cosydydy=du/cosy\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.

 (x2u+3)dx+(2x4u3)cosyducosy=0 Integrating (x2u+3)dx+(2x4u3)du=Cx222ux+3x+2x224ux3u=C3x264x3u+3x=C23x236xsiny3siny+3x=C\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}



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