Question

What is the solution of the differential equation \[{e^{2x - 3y}}dx + {e^{2y - 3x}}dy = 0\]?
A. \[{e^{5x}} + {e^{5y}} = c\]
B. \[{e^{5x}} - {e^{5y}} = c\]
C. \[{e^{5x + 5y}} = c\]
D. None of these

Answer 1

Hint: First we will apply the indices formula to simplify the equation. Then we will separate the variables of the given differential equation. Then we will apply an integration formula to get the solution of the differential equation.

Formula used:
Power formula of indices:
\[{a^{m - n}} = \dfrac{{{a^m}}}{{{a^n}}}\]
\[{a^{m + n}} = {a^m} \cdot {a^n}\]
Integration formula:
\[\int {{e^{mx}}dx} = \dfrac{{{e^{mx}}}}{m} + c\]

Complete step by step solution:
Given differential equation is
\[{e^{2x - 3y}}dx + {e^{2y - 3x}}dy = 0\]
Apply the indices formula:
\[ \Rightarrow \dfrac{{{e^{2x}}}}{{{e^{3y}}}}dx + \dfrac{{{e^{2y}}}}{{{e^{3x}}}}dy = 0\]
Subtract \[\dfrac{{{e^{2y}}}}{{{e^{3x}}}}dy\] from both sides:
\[ \Rightarrow \dfrac{{{e^{2x}}}}{{{e^{3y}}}}dx + \dfrac{{{e^{2y}}}}{{{e^{3x}}}}dy - \dfrac{{{e^{2y}}}}{{{e^{3x}}}}dy = - \dfrac{{{e^{2y}}}}{{{e^{3x}}}}dy\]
\[ \Rightarrow \dfrac{{{e^{2x}}}}{{{e^{3y}}}}dx = - \dfrac{{{e^{2y}}}}{{{e^{3x}}}}dy\]
Multiply \[{e^{3y}} \cdot {e^{3x}}\] on both sides of the equation:
\[ \Rightarrow {e^{2x}} \cdot {e^{3x}}dx = - {e^{2y}} \cdot {e^{3y}}dy\]
Apply indices formula \[{a^{m + n}} = {a^m} \cdot {a^n}\]
\[ \Rightarrow {e^{2x + 3x}}dx = - {e^{2y + 3y}}dy\]
Rewrite the above equation:
\[ \Rightarrow {e^{5x}}dx = - {e^{5y}}dy\]
\[ \Rightarrow {e^{5x}}dx + {e^{5y}}dy = 0\]
Taking integration on both of the equation
\[ \Rightarrow \int {{e^{5x}}dx} + \int {{e^{5y}}dy} = 0\]
Apply integration formula:
\[ \Rightarrow \dfrac{{{e^{5x}}}}{5} + \dfrac{{{e^{5y}}}}{5} = c'\]
Multiply both sides by 5:
\[ \Rightarrow {e^{5x}} + {e^{5y}} = 5c'\]
Now replace \[5c' = c\]
\[ \Rightarrow {e^{5x}} + {e^{5y}} = c\]
Hence option A is the correct option.

Additional information :
The general solution of a differential equation is a solution where we do not have any particular value of integration constant that is C.

Note: Students are often confused with the indices formula. They apply the indices formula on \[{e^{5x}} + {e^{5y}} = c\] and get \[{e^{5x + 5y}} = c\] as a result. But the correct formula is \[{a^{m + n}} = {a^m} \cdot {a^n}\]. Thus the correct solution is \[{e^{5x}} + {e^{5y}} = c\].