Calculus Examples

$y' = 2 x y$ , $y = c e^{x^{2}}$ , $y (0) = 1$

Step 1

Verify that the given solution satisfies the differential equation.

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Step 1.1

Find $y'$ .

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Step 1.1.1

Differentiate both sides of the equation.

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

Step 1.1.2

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

$y'$

Step 1.1.3

Differentiate the right side of the equation.

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Step 1.1.3.1

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

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

Step 1.1.3.2

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 1.1.3.2.1

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

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

Step 1.1.3.2.2

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

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

Step 1.1.3.2.3

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

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

Step 1.1.3.3

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

$c e^{x^{2}} (2 x)$

Step 1.1.3.4

Simplify.

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Step 1.1.3.4.1

Reorder the factors of $c e^{x^{2}} (2 x)$ .

$2 e^{x^{2}} c x$

Step 1.1.3.4.2

Reorder factors in $2 e^{x^{2}} c x$ .

$2 c x e^{x^{2}}$

Step 1.1.4

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

$y' = 2 c x e^{x^{2}}$

Step 1.2

Substitute into the given differential equation.

$2 c x e^{x^{2}} = 2 x (c e^{x^{2}})$

Step 1.3

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

$2 c x e^{x^{2}} = 2 x c e^{x^{2}}$

Step 1.4

The given solution satisfies the given differential equation.

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

Step 2

Substitute in the initial condition.

$1 = c e^{0^{2}}$

Step 3

Solve for $c$ .

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Step 3.1

Rewrite the equation as $c e^{0^{2}} = 1$ .

$c e^{0^{2}} = 1$

Step 3.2

Divide each term in $c e^{0^{2}} = 1$ by $e^{0^{2}}$ and simplify.

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Step 3.2.1

Divide each term in $c e^{0^{2}} = 1$ by $e^{0^{2}}$ .

$\frac{c e^{0^{2}}}{e^{0^{2}}} = \frac{1}{e^{0^{2}}}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $e^{0^{2}}$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{c e^{0^{2}}}{e^{0^{2}}} = \frac{1}{e^{0^{2}}}$

Step 3.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{1}{e^{0^{2}}}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify the denominator.

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Step 3.2.3.1.1

Raising $0$ to any positive power yields $0$ .

$c = \frac{1}{e^{0}}$

Step 3.2.3.1.2

Anything raised to $0$ is $1$ .

$c = \frac{1}{1}$

Step 3.2.3.2

Divide $1$ by $1$ .

$c = 1$

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