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First order integral.

The following first order differential equation is given.

$\\frac{dy}{dx} = \\frac{\\var{a}x^\\var{b-1}}{{x^\\var{b}}+{\\var{c}}}$, where $y=\\var{d}$ when $x=0$

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$\\frac{dy}{dx} = \\frac{\\var{a}x^\\var{b-1}}{{x^\\var{b}}+{\\var{c}}}$, where $y=\\var{d}$ when $x=0$

\\[ \\int{\\frac{\\var{a}x^\\var{b-1}}{{x^\\var{b}}+{\\var{c}}}.dx} \\]

Move constant outside

\\[ \\var{a}\\int{\\frac{x^\\var{b-1}}{{x^\\var{b}}+{\\var{c}}}.dx} \\]

let \\[ u ={x^\\var{b}}+{\\var{c}}\\]

\\[\\frac{du}{dx}=\\var{b} x^\\var{b-1}\\]

\\[\\therefore dx=\\frac{du}{\\var{b} x^\\var{b-1}}\\]

Sub back into equation

\\[ \\var{a}\\int{\\frac{x^\\var{b-1}}{u}.\\frac{du}{\\var{b} x^\\var{b-1}}} \\]

The two x values cancel, leaving

\\[ \\frac{\\var{a}}{\\var{b}}\\int{\\frac{1}{u}.du}=\\frac{\\var{a}}{\\var{b}} ln(u)+C \\]

Sub back in the original denominator

\\[y= \\frac{\\var{a}}{\\var{b}} ln({x^\\var{b}}+{\\var{c}})+C \\]

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Find the general solution for the expression.

Enter values as decimals.

$y=$[[0]]

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The general solution for part a can be expressed as,

$\\frac{\\var{a}}{\\var{b}}ln(x^{\\var{b}}+\\var{c})+C$

Evaluate the constant of integration, C. Input values as decimals.

$C =$ [[0]]

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If the constant of integration is expressed as

\\[C = \\var{Cint} \\]

State the particular solution.

$y =$ [[0]]

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