Question Statement

Solve the following first-order differential equation:

$\frac{d y}{d x} = e^{2 x + 3 y}$

Background and Explanation

This problem involves a separable differential equation. To solve it, we use the properties of exponents to split the function into a product of two separate functions—one containing only $x$ and the other containing only $y$ —allowing us to integrate each side independently.

Solution

Given the differential equation: $\frac{d y}{d x} = e^{2 x + 3 y}$

First, we use the algebraic property of exponents ( $e^{a + b} = e^{a} \cdot e^{b}$ ) to separate the terms on the right side: $\frac{d y}{d x} = e^{2 x} \cdot e^{3 y}$

Next, we perform separation of variables. We move all terms involving $y$ to the left side and all terms involving $x$ to the right side: $\frac{d y}{e ^{3 y}} = e^{2 x} d x$

To make integration easier, we rewrite the left side using a negative exponent: $e^{- 3 y} d y = e^{2 x} d x$

Now, we integrate both sides of the equation: $\int e^{- 3 y} d y = \int e^{2 x} d x$

Using the integration rule $\int e^{a x} d x = \frac{1}{a} e^{a x} + C$ , we get: $\frac{e ^{- 3 y}}{- 3} = \frac{e ^{2 x}}{2} + c_{1}$

This can be rewritten as: $\frac{- 1}{3 e ^{3 y}} = \frac{e ^{2 x}}{2} + c_{1}$

To simplify the form of the general solution, we multiply both sides of the equation by $- 3$ : $\frac{1}{e ^{3 y}} = \frac{- 3}{2} e^{2 x} - 3 c_{1}$

Finally, we can define a new constant $c = - 3 c_{1}$ to represent the arbitrary constant of integration: $\frac{1}{e ^{3 y}} = - \frac{3}{2} e^{2 x} + c$

Key Formulas or Methods Used

Exponent Rules: $e^{a + b} = e^{a} \cdot e^{b}$ and $\frac{1}{e ^{a}} = e^{- a}$
Separation of Variables: A method for solving differential equations by grouping variables.
Exponential Integration Rule: $\int e^{a x} d x = \frac{1}{a} e^{a x} + C$

Summary of Steps

Use exponent laws to separate the $e^{2 x + 3 y}$ term into $e^{2 x} \cdot e^{3 y}$ .
Rearrange the equation so that $y$ terms and $d y$ are on one side, and $x$ terms and $d x$ are on the other.
Integrate both sides with respect to their respective variables.
Simplify the resulting algebraic expression and combine constants.

Solution

Given the differential equation:

\frac{d y}{d x} = e^{2 x + 3 y}

First, we use the algebraic property of exponents (

e^{a + b} = e^{a} \cdot e^{b}

) to separate the terms on the right side:

\frac{d y}{d x} = e^{2 x} \cdot e^{3 y}

Next, we perform separation of variables. We move all terms involving

y

to the left side and all terms involving

x

to the right side:

\frac{d y}{e ^{3 y}} = e^{2 x} d x

To make integration easier, we rewrite the left side using a negative exponent:

e^{- 3 y} d y = e^{2 x} d x

Now, we integrate both sides of the equation:

\int e^{- 3 y} d y = \int e^{2 x} d x

Using the integration rule

\int e^{a x} d x = \frac{1}{a} e^{a x} + C

, we get:

\frac{e ^{- 3 y}}{- 3} = \frac{e ^{2 x}}{2} + c_{1}

This can be rewritten as:

\frac{- 1}{3 e ^{3 y}} = \frac{e ^{2 x}}{2} + c_{1}

To simplify the form of the general solution, we multiply both sides of the equation by

- 3

\frac{1}{e ^{3 y}} = \frac{- 3}{2} e^{2 x} - 3 c_{1}

Finally, we can define a new constant

c = - 3 c_{1}

to represent the arbitrary constant of integration:

\frac{1}{e ^{3 y}} = - \frac{3}{2} e^{2 x} + c

4.2 Q-4

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

4.2 Q-4

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps