Solve the equation.
(dx)/(dt)= (1)/(xe^(t+7x))
Part 1
An implicit solution in the form ​F(t,x)equalsC is 
  
enter your response hereequals​C, where C is an arbitrary constant.
​(Type an expression using t and x as the​ variables.)

To find an implicit solution of the differential equation

\[
\frac{dx}{dt} = \frac{1}{x e^{t + 7x}},
\]

we can follow these steps:

### Step 1: Rearranging the equation

First, multiply both sides by \( x e^{t + 7x} \) to separate the variables:

\[
x e^{t + 7x} \, dx = dt.
\]

### Step 2: Integrating both sides

Next, we integrate both sides. The left-hand side is with respect to \( x \) and the right-hand side is with respect to \( t \):

\[
\int x e^{t + 7x} \, dx = \int dt.
\]

However, the left side is tricky to integrate as is. But we can observe that the exponential term can be treated as a composite function. Let’s substitute for simplicity.

Let:

\[
u = t + 7x.
\]

Then:

\[
du = 7 dx.
\]

Substitute this into the integral and factor out constants where necessary, and continue solving for \( F(t, x) = C \). (We might need a step-by-step approach for handling this substitution.)



Solve the equation. (dx)/(dt)= (1)/(xe^(t+7x)). An implicit solution in the form F(t,x) = C is?

Discover how to solve the differential equation dx/dt = 1/(xe^(t + 7x)) using separation of variables. Learn step-by-step how to find an implicit solution, understand core concepts in differential equations, and apply integration techniques. Suitable for college-level calculus students or advanced high school learners.

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