Calculus Examples

$\frac{d y}{d t} + t y = y$ , $y (1) = 3$

Step 1

Separate the variables.

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Step 1.1

Subtract $t y$ from both sides of the equation.

$\frac{d y}{d t} = y - t y$

Step 1.2

Factor $y$ out of $y - t y$ .

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Step 1.2.1

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

$\frac{d y}{d t} = y^{1} - t y$

Step 1.2.2

Factor $y$ out of $y^{1}$ .

$\frac{d y}{d t} = y \cdot 1 - t y$

Step 1.2.3

Factor $y$ out of $- t y$ .

$\frac{d y}{d t} = y \cdot 1 + y (- t)$

Step 1.2.4

Factor $y$ out of $y \cdot 1 + y (- t)$ .

$\frac{d y}{d t} = y (1 - t)$

Step 1.3

Multiply both sides by $\frac{1}{y}$ .

$\frac{1}{y} \frac{d y}{d t} = \frac{1}{y} (y (1 - t))$

Step 1.4

Cancel the common factor of $y$ .

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Step 1.4.1

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d t} = \frac{1}{y} (y (1 - t))$

Step 1.4.2

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d t} = 1 - t$

Step 1.5

Rewrite the equation.

$\frac{1}{y} d y = (1 - t) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int 1 - t d t$

Step 2.2

The integral of $\frac{1}{y}$ with respect to $y$ is $\ln (| y |)$ .

$\ln (| y |) + C_{1} = \int 1 - t d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$\ln (| y |) + C_{1} = \int d t + \int - t d t$

Step 2.3.2

Apply the constant rule.

$\ln (| y |) + C_{1} = t + C_{2} + \int - t d t$

Step 2.3.3

Since $- 1$ is constant with respect to $t$ , move $- 1$ out of the integral.

$\ln (| y |) + C_{1} = t + C_{2} - \int t d t$

Step 2.3.4

By the Power Rule, the integral of $t$ with respect to $t$ is $\frac{1}{2} t^{2}$ .

$\ln (| y |) + C_{1} = t + C_{2} - (\frac{1}{2} t^{2} + C_{3})$

Step 2.3.5

Simplify.

$\ln (| y |) + C_{1} = t - \frac{1}{2} t^{2} + C_{4}$

Step 2.3.6

Reorder terms.

$\ln (| y |) + C_{1} = - \frac{1}{2} t^{2} + t + C_{4}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$\ln (| y |) = - \frac{1}{2} t^{2} + t + C$

Step 3

Solve for $y$ .

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Step 3.1

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y |)} = e^{- \frac{1}{2} t^{2} + t + C}$

Step 3.2

Rewrite $\ln (| y |) = - \frac{1}{2} t^{2} + t + C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- \frac{1}{2} t^{2} + t + C} = | y |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y | = e^{- \frac{1}{2} t^{2} + t + C}$ .

$| y | = e^{- \frac{1}{2} t^{2} + t + C}$

Step 3.3.2

Combine $t^{2}$ and $\frac{1}{2}$ .

$| y | = e^{- \frac{t^{2}}{2} + t + C}$

Step 3.3.3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y = \pm e^{- \frac{t^{2}}{2} + t + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{- \frac{t^{2}}{2} + t + C}$ as $e^{- \frac{t^{2}}{2} + t} e^{C}$ .

$y = \pm e^{- \frac{t^{2}}{2} + t} e^{C}$

Step 4.2

Reorder $e^{- \frac{t^{2}}{2} + t}$ and $e^{C}$ .

$y = \pm e^{C} e^{- \frac{t^{2}}{2} + t}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{- \frac{t^{2}}{2} + t}$

Step 5

Use the initial condition to find the value of $C$ by substituting $1$ for $t$ and $3$ for $y$ in $y = C e^{- \frac{t^{2}}{2} + t}$ .

$3 = C e^{- \frac{1^{2}}{2} + 1}$

Step 6

Solve for $C$ .

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Step 6.1

Rewrite the equation as $C e^{- \frac{1^{2}}{2} + 1} = 3$ .

$C e^{- \frac{1^{2}}{2} + 1} = 3$

Step 6.2

Simplify.

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Step 6.2.1

One to any power is one.

$C e^{- \frac{1}{2} + 1} = 3$

Step 6.2.2

Write $1$ as a fraction with a common denominator.

$C e^{- \frac{1}{2} + \frac{2}{2}} = 3$

Step 6.2.3

Combine the numerators over the common denominator.

$C e^{\frac{- 1 + 2}{2}} = 3$

Step 6.2.4

Add $- 1$ and $2$ .

$C e^{\frac{1}{2}} = 3$

Step 6.3

Divide each term in $C e^{\frac{1}{2}} = 3$ by $e^{\frac{1}{2}}$ and simplify.

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Step 6.3.1

Divide each term in $C e^{\frac{1}{2}} = 3$ by $e^{\frac{1}{2}}$ .

$\frac{C e^{\frac{1}{2}}}{e^{\frac{1}{2}}} = \frac{3}{e^{\frac{1}{2}}}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor.

$\frac{C e^{\frac{1}{2}}}{e^{\frac{1}{2}}} = \frac{3}{e^{\frac{1}{2}}}$

Step 6.3.2.2

Divide $C$ by $1$ .

$C = \frac{3}{e^{\frac{1}{2}}}$

Step 7

Substitute $\frac{3}{e^{\frac{1}{2}}}$ for $C$ in $y = C e^{- \frac{t^{2}}{2} + t}$ and simplify.

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Step 7.1

Substitute $\frac{3}{e^{\frac{1}{2}}}$ for $C$ .

$y = \frac{3}{e^{\frac{1}{2}}} e^{- \frac{t^{2}}{2} + t}$

Step 7.2

Combine $\frac{3}{e^{\frac{1}{2}}}$ and $e^{- \frac{t^{2}}{2} + t}$ .

$y = \frac{3 e^{- \frac{t^{2}}{2} + t}}{e^{\frac{1}{2}}}$

Step 7.3

Factor $e^{\frac{1}{2}}$ out of $3 e^{- \frac{t^{2}}{2} + t}$ .

$y = \frac{e^{\frac{1}{2}} (3 e^{- \frac{t^{2}}{2} + t - \frac{1}{2}})}{e^{\frac{1}{2}}}$

Step 7.4

Cancel the common factors.

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Step 7.4.1

Multiply by $1$ .

$y = \frac{e^{\frac{1}{2}} (3 e^{- \frac{t^{2}}{2} + t - \frac{1}{2}})}{e^{\frac{1}{2}} \cdot 1}$

Step 7.4.2

Cancel the common factor.

$y = \frac{e^{\frac{1}{2}} (3 e^{- \frac{t^{2}}{2} + t - \frac{1}{2}})}{e^{\frac{1}{2}} \cdot 1}$

Step 7.4.3

Rewrite the expression.

$y = \frac{3 e^{- \frac{t^{2}}{2} + t - \frac{1}{2}}}{1}$

Step 7.4.4

Divide $3 e^{- \frac{t^{2}}{2} + t - \frac{1}{2}}$ by $1$ .

$y = 3 e^{- \frac{t^{2}}{2} + t - \frac{1}{2}}$