# What is the general solution of the differential equation # e^(dy/dx) = x #?

##### 2 Answers

Nov 4, 2017

#### Explanation:

If

then taking the natural log of both sides:

LS

and

RS

Nov 4, 2017

# y = xlnx-x + C #

#### Explanation:

We have:

# e^(dy/dx) = x #

Taking Natural logarithms we have:

# dy/dx = ln x #

Which is a separable Differential Equation, so we can separate the variables to get:

# y = int \ ln \ dx + C #

We can apply integration by Parts

Let

# { (u,=lnx, => (du)/dx,=1/x), ((dv)/dx,=1, => v,=x ) :}#

Then plugging into the IBP formula:

# int \ (u)((dv)/dx) \ dx = (u)(v) - int \ (v)((du)/dx) \ dx #

gives us

# int \ (lnx)(1) \ dx = (lnx)(x) - int \ (x)(1/x) \ dx #

# :. int \ lnx \ dx = xlnx - int 1 \ dx #

# :. int \ lnx \ dx = xlnx - x #

Thus we have the GS:

# y = xlnx-x + C #