Answer:
[tex]x(t) =e^{\frac{1}{3}e^{3x}+9t+C}[/tex]
Step-by-step explanation:
Solve the given differential equation.
[tex]\frac{dx}{dt} = xe^{ 3 t}+9x[/tex]
(1) - Use separation of variables to solve
[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Separable Differential Equation:}}\\\frac{dy}{dx} =f(x)g(y)\\\\\rightarrow\int\frac{dy}{g(y)}=\int f(x)dx \end{array}\right }[/tex]
[tex]\frac{dx}{dt} = xe^{ 3 t}+9x\\\\\Longrightarrow \frac{dx}{dt} = x(e^{ 3 t}+9)\\\\\Longrightarrow \frac{1}{x}dx = (e^{ 3 t}+9)dt\\\\\Longrightarrow \int\frac{1}{x}dx = \int(e^{ 3 t}+9)dt\\\\\Longrightarrow \boxed{\ln(x) =\frac{1}{3}e^{3x}+9t+C}[/tex]
(2) - Simplify to get x(t)
[tex]\ln(x) =\frac{1}{3}e^{3x}+9t+C\\\\\Longrightarrow e^{\ln(x)} =e^{\frac{1}{3}e^{3x}+9t+C}\\\\\therefore \boxed{\boxed{ x(t) =e^{\frac{1}{3}e^{3x}+9t+C}}}[/tex]
Thus, the given DE is solved.
