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

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Type your math problem... (x+1)\frac{dy}{dx}+2y = x

Question

求解微分方程

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

求值

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

两边相乘

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

应用分配属性

(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}

乘以项

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

乘以项

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

乘以项

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

重写表达式

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

由于方程是以标准形式书写的，因此确定函数 P (x) 和 Q (x)

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

将函数 P (x) = \frac{2}{x + 1} 插入积分因子 u (x) 的公式中

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

计算积分

More Steps

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

重写表达式

More Steps

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

将积分因子 u (x) 和函数 Q (x) 代入通解公式

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

计算

More Steps

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

计算

More Steps

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

Solution

More Steps

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

Show Solution