A) \[{{e}^{3x}}+3{{e}^{-4y}}=4\]
B) \[4{{e}^{3x}}-3{{e}^{-4y}}=3\]
C) \[3{{e}^{3x}}+4{{e}^{4y}}=7\]
D) \[4{{e}^{3x}}+3{{e}^{-4y}}=7\]
Correct Answer: D
Solution :
Given differential equation is \[\log \left( \frac{dy}{dx} \right)=3x+4y,\] \[y(0)=0\] \[\Rightarrow \,\,\frac{dy}{dx}={{e}^{3x+4y}}={{e}^{3x}}.{{e}^{4y}}\]\[\Rightarrow \int{{{e}^{-4y}}dy=\int{{{e}^{3x}}dx}}\]\[\Rightarrow \,\,\frac{{{e}^{-4y}}}{-4}=\frac{{{e}^{3x}}}{3}+c\] By using \[y=0\] when \[x=0,\]we get \[c=-\frac{7}{12}\] \[\therefore \] Particular solution is \[4{{e}^{3x}}+3{{e}^{-4y}}=7\]
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