The solution of {e^{2x - 3y}}dx + {e^{2y - 3x | vedclass

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Question

(A) Given differential equation: ${e^{2x - 3y}}dx + {e^{2y - 3x}}dy = 0$Rewrite the terms using exponent rules:${e^{2x}} \cdot {e^{-3y}}dx + {e^{2y}}

Vedclass · Accepted Answer

${e^{5x}} + {e^{5y}} = c$

Vedclass · Answer

(A) Given differential equation: ${e^{2x - 3y}}dx + {e^{2y - 3x}}dy = 0$Rewrite the terms using exponent rules:${e^{2x}} \cdot {e^{-3y}}dx + {e^{2y}} \cdot {e^{-3x}}dy = 0$Multiply the entire equation by ${e^{3x}} \cdot {e^{3y}}$ to separate the variables:${e^{2x}} \cdot {e^{-3y}} \cdot {e^{3x}} \cdot {e^{3y}}dx + {e^{2y}} \cdot {e^{-3x}} \cdot {e^{3x}} \cdot {e^{3y}}dy = 0$Simplify the exponents:${e^{5x}}dx + {e^{5y}}dy = 0$Integrate both sides:$\int {e^{5x}}dx + \int {e^{5y}}dy = \int 0$$\frac{1}{5}{e^{5x}} + \frac{1}{5}{e^{5y}} = C_1$Multiply by $5$ to simplify the constant:${e^{5x}} + {e^{5y}} = 5C_1 = c$Thus,the correct option is $A$.

The solution of ${e^{2x - 3y}}dx + {e^{2y - 3x}}dy = 0$ is

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