Calculus Examples

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

Step 1

Write the problem as a mathematical expression.

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

Step 2

Rewrite the differential equation.

$x y' - y = 2 x^{2} y$

Step 3

Find $y'$ .

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Step 3.1

Differentiate both sides of the equation.

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

Step 3.2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3.3

Differentiate the right side of the equation.

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Step 3.3.1

Since $A$ is constant with respect to $x$ , the derivative of $A x e^{x^{2}}$ with respect to $x$ is $A \frac{d}{d x} [x e^{x^{2}}]$ .

$A \frac{d}{d x} [x e^{x^{2}}]$

Step 3.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = e^{x^{2}}$ .

$A (x \frac{d}{d x} [e^{x^{2}}] + e^{x^{2}} \frac{d}{d x} [x])$

Step 3.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{2}$ .

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Step 3.3.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$A (x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{2}]) + e^{x^{2}} \frac{d}{d x} [x])$

Step 3.3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$A (x (e^{u} \frac{d}{d x} [x^{2}]) + e^{x^{2}} \frac{d}{d x} [x])$

Step 3.3.3.3

Replace all occurrences of $u$ with $x^{2}$ .

$A (x (e^{x^{2}} \frac{d}{d x} [x^{2}]) + e^{x^{2}} \frac{d}{d x} [x])$

Step 3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$A (x (e^{x^{2}} (2 x)) + e^{x^{2}} \frac{d}{d x} [x])$

Step 3.3.5

Raise $x$ to the power of $1$ .

$A (x^{1} x (e^{x^{2}} \cdot (2)) + e^{x^{2}} \frac{d}{d x} [x])$

Step 3.3.6

Raise $x$ to the power of $1$ .

$A (x^{1} x^{1} (e^{x^{2}} \cdot (2)) + e^{x^{2}} \frac{d}{d x} [x])$

Step 3.3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$A (x^{1 + 1} (e^{x^{2}} \cdot (2)) + e^{x^{2}} \frac{d}{d x} [x])$

Step 3.3.8

Simplify the expression.

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Step 3.3.8.1

Add $1$ and $1$ .

$A (x^{2} (e^{x^{2}} \cdot (2)) + e^{x^{2}} \frac{d}{d x} [x])$

Step 3.3.8.2

Move $2$ to the left of $e^{x^{2}}$ .

$A (x^{2} (2 \cdot e^{x^{2}}) + e^{x^{2}} \frac{d}{d x} [x])$

Step 3.3.9

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

$A (x^{2} (2 e^{x^{2}}) + e^{x^{2}} \cdot 1)$

Step 3.3.10

Multiply $e^{x^{2}}$ by $1$ .

$A (x^{2} (2 e^{x^{2}}) + e^{x^{2}})$

Step 3.3.11

Simplify.

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Step 3.3.11.1

Apply the distributive property.

$A (x^{2} (2 e^{x^{2}})) + A e^{x^{2}}$

Step 3.3.11.2

Reorder terms.

$2 e^{x^{2}} x^{2} A + e^{x^{2}} A$

Step 3.3.11.3

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

$2 x^{2} A e^{x^{2}} + A e^{x^{2}}$

Step 3.4

Reform the equation by setting the left side equal to the right side.

$y' = 2 x^{2} A e^{x^{2}} + A e^{x^{2}}$

Step 4

Substitute into the given differential equation.

$x (2 x^{2} A e^{x^{2}} + A e^{x^{2}}) - (A x e^{x^{2}}) = 2 x^{2} (A x e^{x^{2}})$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Apply the distributive property.

$x (2 x^{2} A e^{x^{2}}) + x (A e^{x^{2}}) - (A x e^{x^{2}}) = 2 x^{2} (A x e^{x^{2}})$

Step 5.1.2

Rewrite using the commutative property of multiplication.

$2 x (x^{2} A e^{x^{2}}) + x (A e^{x^{2}}) - (A x e^{x^{2}}) = 2 x^{2} (A x e^{x^{2}})$

Step 5.1.3

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.1.3.1

Move $x^{2}$ .

$2 (x^{2} x) (A e^{x^{2}}) + x (A e^{x^{2}}) - (A x e^{x^{2}}) = 2 x^{2} (A x e^{x^{2}})$

Step 5.1.3.2

Multiply $x^{2}$ by $x$ .

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Step 5.1.3.2.1

Raise $x$ to the power of $1$ .

$2 (x^{2} x^{1}) (A e^{x^{2}}) + x (A e^{x^{2}}) - (A x e^{x^{2}}) = 2 x^{2} (A x e^{x^{2}})$

Step 5.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{2 + 1} (A e^{x^{2}}) + x (A e^{x^{2}}) - (A x e^{x^{2}}) = 2 x^{2} (A x e^{x^{2}})$

Step 5.1.3.3

Add $2$ and $1$ .

$2 x^{3} (A e^{x^{2}}) + x (A e^{x^{2}}) - (A x e^{x^{2}}) = 2 x^{2} (A x e^{x^{2}})$

$2 x^{3} (A e^{x^{2}}) + x (A e^{x^{2}}) - A x e^{x^{2}} = 2 x^{2} (A x e^{x^{2}})$

Step 5.2

Combine the opposite terms in $2 x^{3} (A e^{x^{2}}) + x (A e^{x^{2}}) - A x e^{x^{2}}$ .

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Step 5.2.1

Reorder the factors in the terms $x (A e^{x^{2}})$ and $- A x e^{x^{2}}$ .

$2 x^{3} (A e^{x^{2}}) + e^{x^{2}} A x - e^{x^{2}} A x = 2 x^{2} (A x e^{x^{2}})$

Step 5.2.2

Subtract $e^{x^{2}} A x$ from $e^{x^{2}} A x$ .

$2 x^{3} (A e^{x^{2}}) + 0 = 2 x^{2} (A x e^{x^{2}})$

Step 5.2.3

Add $2 x^{3} (A e^{x^{2}})$ and $0$ .

$2 x^{3} (A e^{x^{2}}) = 2 x^{2} (A x e^{x^{2}})$

Step 5.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 5.3.1

Move $x$ .

$2 x^{3} (A e^{x^{2}}) = 2 (x \cdot x^{2}) (A e^{x^{2}})$

Step 5.3.2

Multiply $x$ by $x^{2}$ .

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Step 5.3.2.1

Raise $x$ to the power of $1$ .

$2 x^{3} (A e^{x^{2}}) = 2 (x^{1} x^{2}) (A e^{x^{2}})$

Step 5.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{3} (A e^{x^{2}}) = 2 x^{1 + 2} (A e^{x^{2}})$

Step 5.3.3

Add $1$ and $2$ .

$2 x^{3} (A e^{x^{2}}) = 2 x^{3} (A e^{x^{2}})$

Step 5.4

Reorder factors in $2 x^{3} (A e^{x^{2}}) = 2 x^{3} (A e^{x^{2}})$ .

$2 x^{3} A e^{x^{2}} = 2 x^{3} A e^{x^{2}}$

Step 6

The given solution satisfies the given differential equation.

$y = A x e^{x^{2}}$ is a solution to $x y' - y = 2 x^{2} y$