Calculus Examples

$e^{x} \frac{d y}{d x} + 3 y = x^{2} y$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Reorder factors in $e^{x} \frac{d y}{d x} + 3 y$ .

$\frac{d y}{d x} e^{x} + 3 y = x^{2} y$

Step 1.1.2

Subtract $3 y$ from both sides of the equation.

$\frac{d y}{d x} e^{x} = x^{2} y - 3 y$

Step 1.1.3

Divide each term in $\frac{d y}{d x} e^{x} = x^{2} y - 3 y$ by $e^{x}$ and simplify.

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Step 1.1.3.1

Divide each term in $\frac{d y}{d x} e^{x} = x^{2} y - 3 y$ by $e^{x}$ .

$\frac{\frac{d y}{d x} e^{x}}{e^{x}} = \frac{x^{2} y}{e^{x}} + \frac{- 3 y}{e^{x}}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $e^{x}$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

$\frac{\frac{d y}{d x} e^{x}}{e^{x}} = \frac{x^{2} y}{e^{x}} + \frac{- 3 y}{e^{x}}$

Step 1.1.3.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{x^{2} y}{e^{x}} + \frac{- 3 y}{e^{x}}$

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Move the negative in front of the fraction.

$\frac{d y}{d x} = \frac{x^{2} y}{e^{x}} - \frac{3 y}{e^{x}}$

Step 1.2

Factor.

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Step 1.2.1

Combine the numerators over the common denominator.

$\frac{d y}{d x} = \frac{x^{2} y - 3 y}{e^{x}}$

Step 1.2.2

Factor $y$ out of $x^{2} y - 3 y$ .

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Step 1.2.2.1

Factor $y$ out of $x^{2} y$ .

$\frac{d y}{d x} = \frac{y x^{2} - 3 y}{e^{x}}$

Step 1.2.2.2

Factor $y$ out of $- 3 y$ .

$\frac{d y}{d x} = \frac{y x^{2} + y \cdot - 3}{e^{x}}$

Step 1.2.2.3

Factor $y$ out of $y x^{2} + y \cdot - 3$ .

$\frac{d y}{d x} = \frac{y (x^{2} - 3)}{e^{x}}$

Step 1.3

Regroup factors.

$\frac{d y}{d x} = y \frac{x^{2} - 3}{e^{x}}$

Step 1.4

Multiply both sides by $\frac{1}{y}$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y \frac{x^{2} - 3}{e^{x}})$

Step 1.5

Cancel the common factor of $y$ .

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Step 1.5.1

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y \frac{x^{2} - 3}{e^{x}})$

Step 1.5.2

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = \frac{x^{2} - 3}{e^{x}}$

Step 1.6

Rewrite the equation.

$\frac{1}{y} d y = \frac{x^{2} - 3}{e^{x}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int \frac{x^{2} - 3}{e^{x}} d x$

Step 2.2

The integral of $\frac{1}{y}$ with respect to $y$ is $\ln (| y |)$ .

$\ln (| y |) + C_{1} = \int \frac{x^{2} - 3}{e^{x}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Simplify the expression.

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Step 2.3.1.1

Negate the exponent of $e^{x}$ and move it out of the denominator.

$\ln (| y |) + C_{1} = \int (x^{2} - 3) {(e^{x})}^{- 1} d x$

Step 2.3.1.2

Multiply the exponents in ${(e^{x})}^{- 1}$ .

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Step 2.3.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (| y |) + C_{1} = \int (x^{2} - 3) e^{x \cdot - 1} d x$

Step 2.3.1.2.2

Move $- 1$ to the left of $x$ .

$\ln (| y |) + C_{1} = \int (x^{2} - 3) e^{- 1 \cdot x} d x$

Step 2.3.1.2.3

Rewrite $- 1 x$ as $- x$ .

$\ln (| y |) + C_{1} = \int (x^{2} - 3) e^{- x} d x$

Step 2.3.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2} - 3$ and $d v = e^{- x}$ .

$\ln (| y |) + C_{1} = (x^{2} - 3) (- e^{- x}) - \int - e^{- x} (2 x) d x$

Step 2.3.3

Multiply $2$ by $- 1$ .

$\ln (| y |) + C_{1} = (x^{2} - 3) (- e^{- x}) - \int - 2 e^{- x} x d x$

Step 2.3.4

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$\ln (| y |) + C_{1} = (x^{2} - 3) (- e^{- x}) - (- 2 \int e^{- x} x d x)$

Step 2.3.5

Multiply $- 2$ by $- 1$ .

$\ln (| y |) + C_{1} = (x^{2} - 3) (- e^{- x}) + 2 \int e^{- x} x d x$

Step 2.3.6

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{- x}$ .

$\ln (| y |) + C_{1} = (x^{2} - 3) (- e^{- x}) + 2 (x (- e^{- x}) - \int - e^{- x} d x)$

Step 2.3.7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\ln (| y |) + C_{1} = (x^{2} - 3) (- e^{- x}) + 2 (x (- e^{- x}) - - \int e^{- x} d x)$

Step 2.3.8

Simplify.

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Step 2.3.8.1

Multiply $- 1$ by $- 1$ .

$\ln (| y |) + C_{1} = (x^{2} - 3) (- e^{- x}) + 2 (x (- e^{- x}) + 1 \int e^{- x} d x)$

Step 2.3.8.2

Multiply $\int e^{- x} d x$ by $1$ .

$\ln (| y |) + C_{1} = (x^{2} - 3) (- e^{- x}) + 2 (x (- e^{- x}) + \int e^{- x} d x)$

Step 2.3.9

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.9.1

Let $u = - x$ . Find $\frac{d u}{d x}$ .

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Step 2.3.9.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 2.3.9.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x]$

Step 2.3.9.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 2.3.9.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 2.3.9.2

Rewrite the problem using $u$ and $d u$ .

$\ln (| y |) + C_{1} = (x^{2} - 3) (- e^{- x}) + 2 (x (- e^{- x}) + \int - e^{u} d u)$

Step 2.3.10

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$\ln (| y |) + C_{1} = (x^{2} - 3) (- e^{- x}) + 2 (x (- e^{- x}) - \int e^{u} d u)$

Step 2.3.11

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$\ln (| y |) + C_{1} = (x^{2} - 3) (- e^{- x}) + 2 (x (- e^{- x}) - (e^{u} + C_{2}))$

Step 2.3.12

Rewrite $(x^{2} - 3) (- e^{- x}) + 2 (x (- e^{- x}) - (e^{u} + C_{2}))$ as $- x^{2} e^{- x} + 3 e^{- x} + 2 (x (- e^{- x}) - e^{u}) + C_{2}$ .

$\ln (| y |) + C_{1} = - x^{2} e^{- x} + 3 e^{- x} + 2 (x (- e^{- x}) - e^{u}) + C_{2}$

Step 2.3.13

Replace all occurrences of $u$ with $- x$ .

$\ln (| y |) + C_{1} = - x^{2} e^{- x} + 3 e^{- x} + 2 (x (- e^{- x}) - e^{- x}) + C_{2}$

Step 2.3.14

Reorder terms.

$\ln (| y |) + C_{1} = - e^{- x} x^{2} + 3 e^{- x} + 2 (x (- e^{- x}) - e^{- x}) + C_{2}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$\ln (| y |) = - e^{- x} x^{2} + 3 e^{- x} + 2 (x (- e^{- x}) - e^{- x}) + C$

Step 3

Solve for $y$ .

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Step 3.1

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y |)} = e^{- e^{- x} x^{2} + 3 e^{- x} + 2 (x (- e^{- x}) - e^{- x}) + C}$

Step 3.2

Rewrite $\ln (| y |) = - e^{- x} x^{2} + 3 e^{- x} + 2 (x (- e^{- x}) - e^{- x}) + C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- e^{- x} x^{2} + 3 e^{- x} + 2 (x (- e^{- x}) - e^{- x}) + C} = | y |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y | = e^{- e^{- x} x^{2} + 3 e^{- x} + 2 (x (- e^{- x}) - e^{- x}) + C}$ .

$| y | = e^{- e^{- x} x^{2} + 3 e^{- x} + 2 (x (- e^{- x}) - e^{- x}) + C}$

Step 3.3.2

Simplify $e^{- e^{- x} x^{2} + 3 e^{- x} + 2 (x (- e^{- x}) - e^{- x}) + C}$ .

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Rewrite using the commutative property of multiplication.

$| y | = e^{- e^{- x} x^{2} + 3 e^{- x} + 2 (- x e^{- x} - e^{- x}) + C}$

Step 3.3.2.1.2

Apply the distributive property.

$| y | = e^{- e^{- x} x^{2} + 3 e^{- x} + 2 (- x e^{- x}) + 2 (- e^{- x}) + C}$

Step 3.3.2.1.3

Multiply $- 1$ by $2$ .

$| y | = e^{- e^{- x} x^{2} + 3 e^{- x} - 2 (x e^{- x}) + 2 (- e^{- x}) + C}$

Step 3.3.2.1.4

Multiply $- 1$ by $2$ .

$| y | = e^{- e^{- x} x^{2} + 3 e^{- x} - 2 x e^{- x} - 2 e^{- x} + C}$

Step 3.3.2.2

Simplify by adding terms.

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Step 3.3.2.2.1

Subtract $2 e^{- x}$ from $3 e^{- x}$ .

$| y | = e^{- e^{- x} x^{2} - 2 x e^{- x} + e^{- x} + C}$

Step 3.3.2.2.2

Reorder factors in $e^{- e^{- x} x^{2} - 2 x e^{- x} + e^{- x} + C}$ .

$| y | = e^{- x^{2} e^{- x} - 2 x e^{- x} + e^{- x} + C}$

Step 3.3.3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y = \pm e^{- x^{2} e^{- x} - 2 x e^{- x} + e^{- x} + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{- x^{2} e^{- x} - 2 x e^{- x} + e^{- x} + C}$ as $e^{- x^{2} e^{- x} - 2 x e^{- x} + e^{- x}} e^{C}$ .

$y = \pm e^{- x^{2} e^{- x} - 2 x e^{- x} + e^{- x}} e^{C}$

Step 4.2

Reorder $e^{- x^{2} e^{- x} - 2 x e^{- x} + e^{- x}}$ and $e^{C}$ .

$y = \pm e^{C} e^{- x^{2} e^{- x} - 2 x e^{- x} + e^{- x}}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{- x^{2} e^{- x} - 2 x e^{- x} + e^{- x}}$