What is the solution for ((x+1)dy)/(dx)=x+6 ?

The solution for ((x+1)dy)/(dx)=x+6 is y=5ln(x+1)+x+c_{1}

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((x+1)dy)/(dx)=x+6

\frac{( x + 1 ) d y}{d x} = x + 6

y = 5 ln (x + 1) + x + c_{1}

Solution steps

\frac{( x + 1 ) d y}{d x} = x + 6

Let

y

be the dependent variable. Divide by

d x

\frac{( x + 1 ) \frac{d y}{d x}}{1} = x + 6

Substitute

\frac{d y}{d x}

with

y^{'} (x)

\frac{( x + 1 ) y ^{^{'}} ( x )}{1} = x + 6

Isolate

y^{'} (x) : y^{'} (x) = \frac{x + 6}{x + 1}

f^{^{'}} (x) = g (x)

then

f (x) = \int g (x) d x

y = \int \frac{x + 6}{x + 1} d x

\int \frac{x + 6}{x + 1} d x = 5 ln (x + 1) + x + c_{1}

y = 5 ln (x + 1) + x + c_{1}

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\frac{d}{d x} (- 1 + e^{2 x})

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What is the solution for ((x+1)dy)/(dx)=x+6 ?
The solution for ((x+1)dy)/(dx)=x+6 is y=5ln(x+1)+x+c_{1}