Solve (x+1)\frac{dy}{dx}+2y = x

Question

Solve the differential equation

y = \frac{2 x ^{3} + 3 x ^{2} + C}{6 x ^{2} + 12 x + 6}, C \in R

Evaluate

(x + 1) \frac{d y}{d x} + 2 y = x

Multiply both sides

((x + 1) \frac{d y}{d x} + 2 y) \times \frac{1}{x + 1} = x \times \frac{1}{x + 1}

Apply the distributive property

(x + 1) \frac{d y}{d x} \times \frac{1}{x + 1} + 2 y \times \frac{1}{x + 1} = x \times \frac{1}{x + 1}

Multiply the terms

\frac{d y}{d x} + 2 y \times \frac{1}{x + 1} = x \times \frac{1}{x + 1}

Multiply the terms

\frac{d y}{d x} + \frac{2 y}{x + 1} = x \times \frac{1}{x + 1}

Multiply the terms

\frac{d y}{d x} + \frac{2 y}{x + 1} = \frac{x}{x + 1}

Rewrite the expression

\frac{d y}{d x} + \frac{2}{x + 1} \times y = \frac{x}{x + 1}

Since the equation is written in standard form, determine the functions P (x) and Q (x)

P (x) = \frac{2}{x + 1} Q (x) = \frac{x}{x + 1}

Insert the function P (x) = \frac{2}{x + 1} into the formula for the integrating factor u (x)

u (x) = e^{\int \frac{2}{x + 1} d x} Q (x) = \frac{x}{x + 1}

Evaluate the integral

More Steps

u (x) = e^{2 l n (x + 1)} Q (x) = \frac{x}{x + 1}

Rewrite the expression

More Steps

u (x) = (x + 1)^{2} Q (x) = \frac{x}{x + 1}

Insert the integrating factor u (x) and the function Q (x) into the general solution formula

y = \frac{1}{( x + 1 ) ^{2}} \times \int \frac{x}{x + 1} \times (x + 1)^{2} d x

Calculate

More Steps

y = \frac{1}{( x + 1 ) ^{2}} \times (\frac{x ^{3}}{3} + \frac{x ^{2}}{2} + C), C \in R

Calculate

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y = \frac{2 x ^{3} + 3 x ^{2} + C}{6 ( x + 1 ) ^{2}}, C \in R

Solution

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y = \frac{2 x ^{3} + 3 x ^{2} + C}{6 x ^{2} + 12 x + 6}, C \in R

Show Solution