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The solution to the given differential equation [tex]\frac{dy}{dx} = e^{4x+5y}[/tex] is , [tex]\frac{e^{-5y}}{-5}=\frac{e^{4x}}{4} +c[/tex], where c is constant of integration.
For given question,
We have been given a differential equation [tex]\frac{dy}{dx} = e^{4x+5y}[/tex]
We know that for any real number a, m, n,
[tex]a^{m + n} = a^m \times a^n[/tex]
⇒ dy/dx = [tex]e^{4x}[/tex] × [tex]e^{5y}[/tex]
Separating the variables (x and its differential in one side and y and its differential in another side )
⇒ [tex]\frac{1}{e^{5y}}[/tex] dy = [tex]e^{4x}[/tex] dx
⇒ [tex]e^{-5y}[/tex] dy = [tex]e^{4x}[/tex] dx
Integrating on both the sides,
⇒ [tex]\int e^{-5y}[/tex] dy = [tex]\int e^{4x}[/tex] dx
We know that, [tex]\int e^{ax}\, dx=\frac{e^{ax}}{a} +C[/tex]
⇒ [tex]\int e^{4x}\, dx=\frac{e^{4x}}{4} +C[/tex]
and [tex]\int e^{-5y}\, dy=\frac{e^{-5y}}{-5} +C[/tex]
So the solution is, [tex]\frac{e^{-5y}}{-5}=\frac{e^{4x}}{4} +c[/tex], where c is constant of integration.
Therefore, the solution to the given differential equation [tex]\frac{dy}{dx} = e^{4x+5y}[/tex] is , [tex]\frac{e^{-5y}}{-5}=\frac{e^{4x}}{4} +c[/tex], where c is constant of integration.
Learn more about the differential equation here:
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