Calculus Examples

$y' = y - 2 x$ ,

$y = 3 x$

Step 1

Find

$y'$ .

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (3 x)$

Step 1.2

The derivative of

$y$ with respect to

$x$ is

$y'$ .

$y'$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

Since

$3$ is constant with respect to

$x$ , the derivative of

$3 x$ with respect to

$x$ is

$3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 1.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$3 \cdot 1$

Step 1.3.3

Multiply

$3$ by

$1$ .

$3$

Step 1.4

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

$y' = 3$

Step 2

Substitute into the given differential equation.

$3 = 3 x - 2 x$

Step 3

Subtract

$2 x$ from

$3 x$ .

$3 = x$

Step 4

The given solution does not satisfy the given differential equation.

$y = 3 x$ is not a solution to

$y' = y - 2 x$

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