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

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

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

(x + 1) d y = (x + 6) d x

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

Solution steps

(x + 1) d y = (x + 6) d x

Let

y

be the dependent variable. Divide by

d x

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

Substitute

\frac{d y}{d x}

with

y^{'} (x)

(x + 1) y^{^{'}} (x) = 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}

\frac{\partial}{\partial x} (x^{2} + x)

y^{^{''}} - 2 y^{^{'}} + y = 0, y (0) = 2, y^{^{'}} (0) = 6

y = x^{2}, y = x^{4}

\int \frac{x + 4}{x ^{2} + 2 x + 5} d x

\frac{d}{d x} (\frac{- x ^{2} - 7}{( x ^{2} - 7 ) ^{2}})

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