# First-Order Nonlinear ODE from transient heat transfer

1. Jun 26, 2012

### mafra

A problem from a heat transfer book with conduction and radiation led me to a differential equation like this:

T'(t) = a - b*T(t) - c*T(t)^4

Although my professor said that there wouldn't be an analytical solution for this one and to get the answer by an iterative method I got curious and tried Wolfram Alpha. For my surprise it gave a solution, but I just can't understand the answer and neither the steps for solving it

Here it is:
http://www.wolframalpha.com/input/?i=T'(t)+=+a+-+b*T(t)+-+c*T(t)^4

Could anyone give me some light here?

2. Jun 27, 2012

### JJacquelin

The solution is shown in attachment.
WolframAlpha gives a closed form of the integral in terms of the four roots of the polynomial :
a - b*T - c*T^4 = c*(T-w1)*(T-w2)*(T-w3)*(T-w4)

#### Attached Files:

• ###### Integral.JPG
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2.4 KB
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3. Jun 27, 2012

### mafra

Thanks mate! That's simpler than I thought. So problably he was referring to an analytical solution for T(t)