Question

Solve the equation. dx/dt =3/xet +9x An implicit solution in the form F(t.x)C, where C is an arbitrary constant.

Answer 1

Answer:

$\text{The implicit solution:} \frac{1}{81} e^{9x}(9x - 1) + \frac{3}{e^t} = C$

Step-by-step explanation:

It is given that there is arbitrary constant C and we have to find the implicit solution. Therefore, first separate the variable that is given in equation and then use integration to find the implicit solution. Here, below is the calculation.

 The given equation is:

$\frac{dx}{dt} = \frac{3}{xe^{t-9x}}$

Now, if we use separation of variable.

$\frac{dx}{dt} = \frac{3}{xe^{t-9x}} \\\frac{dx}{dt} = \frac{3}{xe^{9x}e^{t}} \\xe^{9x}dx = \frac{3}{e^{t}}dt$

Now integrate both side:

$\int xe^{9x} dx = \int \frac{3}{e^{t}} dt \\\frac{e^{9x}}{9}(x) - \int \left [ \frac{e^{9x}}{9} \right]dx = -3e^{-t} + C$

$\frac{xe^{9x}}{9} - \frac{e^{9x}}{81} = -3e^{-t} + C \\\frac{1}{81} e^{9x}(9x - 1) + \frac{3}{e^t} = C$

Thus, the implicit solution is:

$\frac{1}{81} e^{9x}(9x - 1) + \frac{3}{e^t} = C$