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Question :

y'=x^2,y(0)=2

求解微分方程

y = \frac{x ^{3} + 6}{3}

求值

y^{'} = x^{2}, y (0) = 2

忽略初始条件

y^{'} = x^{2}

重写表达式

\frac{d y}{d x} = x^{2}

变换表达式

d y = x^{2} d x

对等式左侧的 y 和等式右侧的 x 进行积分

\int 1 d y = \int x^{2} d x

计算

More Steps

求值

\int 1 d y

使用积分 \int k d x = k x 的属性

y

添加积分常数 C_{1}

y + C_{1}, C_{1} \in R

y + C_{1} = \int x^{2} d x, C_{1} \in R

计算

More Steps

求值

\int x^{2} d x

使用积分 \int x^{n} d x = \frac{x ^{n + 1}}{n + 1} 的属性

\frac{x ^{2 + 1}}{2 + 1}

把这些数相加

\frac{x ^{3}}{2 + 1}

把这些数相加

\frac{x ^{3}}{3}

添加积分常数 C_{2}

\frac{x ^{3}}{3} + C_{2}, C_{2} \in R

y + C_{1} = \frac{x ^{3}}{3} + C_{2}, C_{1} \in R, C_{2} \in R

由于积分常数 C_{1} 和 C_{2} 是任意常数，因此将它们替换为常数 C

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

使用初始条件 y (0) = 2 将 0 替换为 x ，将 2 替换为 y

2 = \frac{0 ^{3}}{3} + C

计算

More Steps

求值

\frac{0 ^{3}}{3} + C

计算

\frac{0}{3} + C

把各项相除

0 + C

删除 0 不会更改值，因此将其从表达式中删除

C

2 = C

交换方程两边

C = 2

要找到特定的解决方案，请在通用解决方案 y = \frac{x ^{3}}{3} + C 中用 2 替换 C

y = \frac{x ^{3}}{3} + 2

Solution

More Steps

求值

\frac{x ^{3}}{3} + 2

将分数减少到一个公分母

\frac{x ^{3}}{3} + \frac{2 \times 3}{3}

将所有分子写在公分母上方

\frac{x ^{3} + 2 \times 3}{3}

乘以数字

\frac{x ^{3} + 6}{3}

y = \frac{x ^{3} + 6}{3}

Show Solution

Frequently Asked Questions

使用变量分离求解 \(y'=x^2,y(0)=2\)

$y = \frac{x ^{3} + 6}{3}$

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