Calculus Examples

$y' = 2 y$ , $y = c e^{2 x}$ , $y (0) = 3$

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.

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

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 $c e^{2 x}$ with respect to $x$ is $c \frac{d}{d x} [e^{2 x}]$ .

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

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 $g (x) = 2 x$ .

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Step 1.1.3.2.1

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

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

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 $a$ = $e$ .

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

Step 1.1.3.2.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 1.1.3.3

Differentiate.

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Step 1.1.3.3.1

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

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

Step 1.1.3.3.2

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

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

Step 1.1.3.3.3

Simplify the expression.

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Step 1.1.3.3.3.1

Multiply $2$ by $1$ .

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

Step 1.1.3.3.3.2

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

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

Step 1.1.3.4

Simplify.

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Step 1.1.3.4.1

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

$2 e^{2 x} c$

Step 1.1.3.4.2

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

$2 c e^{2 x}$

Step 1.1.4

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

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

Step 1.2

Substitute into the given differential equation.

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

Step 1.3

Remove parentheses.

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

Step 1.4

The given solution satisfies the given differential equation.

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

Step 2

Substitute in the initial condition.

$3 = c e^{2 \cdot 0}$

Step 3

Solve for $c$ .

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Step 3.1

Rewrite the equation as $c e^{2 \cdot 0} = 3$ .

$c e^{2 \cdot 0} = 3$

Step 3.2

Simplify $c e^{2 \cdot 0}$ .

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Step 3.2.1

Multiply $2$ by $0$ .

$c e^{0} = 3$

Step 3.2.2

Anything raised to $0$ is $1$ .

$c \cdot 1 = 3$

Step 3.2.3

Multiply $c$ by $1$ .

$c = 3$

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