how to solve this differential equation:

$A\cdot(dT(x)/dx)(1873.382+2.2111\cdot T(x))=90457.5-2.149\cdot10^{-10 }-10\cdot T(x)^4$ where A is a constant

Thank you


It looks like your equation takes the form

$$A \frac{dT}{dx} (B + C T) = D - E T^4$$

where the constants are all positive. Then this equation may be turned around to produce

$$\int dT \frac{B+C T}{D-ET^4} = \frac{X}{A} + \text{constant}$$

A good way to attack that integral is to note that

$$(D-E T^4) = (\sqrt{D}-\sqrt{E} T^2) (\sqrt{D}+\sqrt{E} T^2)$$

and then use partial fractions:

$$\frac{1}{D-E T^4} = \frac{1}{2 \sqrt{D}}\left(\frac{1}{\sqrt{D}-\sqrt{E} T^2} + \frac{1}{\sqrt{D}+\sqrt{E} T^2}\right)$$

Using substitution, you may see that the resulting integral is a sum of terms involving arctangents and logarithms. This then gives you $x(T)$, which you would need to invert to get $T(x)$.


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