Calculus Examples Differential Equations

Verify the Differential Equation Solution

$y' = \frac{3 y}{x}$ , $y = 3 x$

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.

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Divide $3 \cdot (3)$ by $1$ .

$3 = 3 \cdot (3)$

Step 3.2

Multiply $3$ by $3$ .

$3 = 9$

Step 4

The given solution does not satisfy the given differential equation.

$y = 3 x$ is not a solution to $y' = \frac{3 y}{x}$

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Verify the Differential Equation Solution

Differential Equations

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