# What is a particular solution to the differential equation #dy/dx=(4sqrtylnx)/x# with #y(e)=1#?

##### 1 Answer

Feb 11, 2017

#### Explanation:

We have:

# dy/dx = (4sqrt(y)lnx)/x #

Which is a first order linear separable Differential Equation, so we can rearrange to get:

# 1/sqrt(y) dy/dx = (4lnx)/x #

and separate the variables to get:

# int \ y^(-1/2) \ dy = int \ (4lnx)/x \ dx #

And then we can integrate to get:

# y^(1/2)/(1/2) = (4)(ln^2x/2) + C #

# :. 2sqrt(y) = 2ln^2x + C #

Using

# :. 2 = 2ln^2e + C #

# :. C=0 #

Hence the particular solution is:

# \ \ \ 2sqrt(y) = 2ln^2x #

# :. sqrt(y) = ln^2x #

# :. \ \ \ y = ln^4x #

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