Solved on Mar 08, 2024

Find the solution to the differential equation dydx=e2x+2y\frac{d y}{d x}=e^{2 x}+2 y. Options: A. y=e2xy=e^{2 x}, B. y=e2xy=-e^{2 x}, C. y=xe2xy=x e^{2 x}, D. y=xe2xy=-x e^{2 x}.

STEP 1

Assumptions
1. We are given the differential equation dydx=e2x+2y\frac{d y}{d x}=e^{2 x}+2 y.
2. We need to find which of the given options is a solution to the differential equation.

STEP 2

To verify which of the given options is a solution, we will take the derivative of each option with respect to xx and check if it satisfies the differential equation.

STEP 3

Start by taking the derivative of option A, y=e2xy=e^{2 x}.
dydx=ddx(e2x)\frac{d y}{d x} = \frac{d}{d x}(e^{2 x})

STEP 4

Apply the chain rule to differentiate e2xe^{2 x}.
dydx=2e2x\frac{d y}{d x} = 2e^{2 x}

STEP 5

Now, substitute y=e2xy=e^{2 x} and dydx=2e2x\frac{d y}{d x}=2e^{2 x} into the differential equation to check for consistency.
2e2x=?e2x+2(e2x)2e^{2 x} \stackrel{?}{=} e^{2 x} + 2(e^{2 x})

STEP 6

Simplify the right-hand side of the equation.
2e2x=?e2x+2e2x2e^{2 x} \stackrel{?}{=} e^{2 x} + 2e^{2 x}
2e2x=?3e2x2e^{2 x} \stackrel{?}{=} 3e^{2 x}

STEP 7

Since 2e2x3e2x2e^{2 x} \neq 3e^{2 x}, option A is not a solution to the differential equation.

STEP 8

Next, take the derivative of option B, y=e2xy=-e^{2 x}.
dydx=ddx(e2x)\frac{d y}{d x} = \frac{d}{d x}(-e^{2 x})

STEP 9

Apply the chain rule to differentiate e2x-e^{2 x}.
dydx=2e2x\frac{d y}{d x} = -2e^{2 x}

STEP 10

Now, substitute y=e2xy=-e^{2 x} and dydx=2e2x\frac{d y}{d x}=-2e^{2 x} into the differential equation to check for consistency.
2e2x=?e2x+2(e2x)-2e^{2 x} \stackrel{?}{=} e^{2 x} + 2(-e^{2 x})

STEP 11

Simplify the right-hand side of the equation.
2e2x=?e2x2e2x-2e^{2 x} \stackrel{?}{=} e^{2 x} - 2e^{2 x}
2e2x=?e2x-2e^{2 x} \stackrel{?}{=} -e^{2 x}

STEP 12

Since 2e2xe2x-2e^{2 x} \neq -e^{2 x}, option B is not a solution to the differential equation.

STEP 13

Now, take the derivative of option C, y=xe2xy=x e^{2 x}.
dydx=ddx(xe2x)\frac{d y}{d x} = \frac{d}{d x}(x e^{2 x})

STEP 14

Apply the product rule to differentiate xe2xx e^{2 x}.
dydx=e2x+2xe2x\frac{d y}{d x} = e^{2 x} + 2x e^{2 x}

STEP 15

Now, substitute y=xe2xy=x e^{2 x} and dydx=e2x+2xe2x\frac{d y}{d x}=e^{2 x} + 2x e^{2 x} into the differential equation to check for consistency.
e2x+2xe2x=?e2x+2(xe2x)e^{2 x} + 2x e^{2 x} \stackrel{?}{=} e^{2 x} + 2(x e^{2 x})

STEP 16

Simplify the right-hand side of the equation.
e2x+2xe2x=?e2x+2xe2xe^{2 x} + 2x e^{2 x} \stackrel{?}{=} e^{2 x} + 2x e^{2 x}

STEP 17

Since e2x+2xe2x=e2x+2xe2xe^{2 x} + 2x e^{2 x} = e^{2 x} + 2x e^{2 x}, option C is a solution to the differential equation.

STEP 18

Although we have found a solution, we should still check option D for completeness.
Take the derivative of option D, y=xe2xy=-x e^{2 x}.
dydx=ddx(xe2x)\frac{d y}{d x} = \frac{d}{d x}(-x e^{2 x})

STEP 19

Apply the product rule to differentiate xe2x-x e^{2 x}.
dydx=e2x2xe2x\frac{d y}{d x} = -e^{2 x} - 2x e^{2 x}

STEP 20

Now, substitute y=xe2xy=-x e^{2 x} and dydx=e2x2xe2x\frac{d y}{d x}=-e^{2 x} - 2x e^{2 x} into the differential equation to check for consistency.
e2x2xe2x=?e2x+2(xe2x)-e^{2 x} - 2x e^{2 x} \stackrel{?}{=} e^{2 x} + 2(-x e^{2 x})

STEP 21

Simplify the right-hand side of the equation.
e2x2xe2x=?e2x2xe2x-e^{2 x} - 2x e^{2 x} \stackrel{?}{=} e^{2 x} - 2x e^{2 x}

STEP 22

Since e2x2xe2xe2x2xe2x-e^{2 x} - 2x e^{2 x} \neq e^{2 x} - 2x e^{2 x}, option D is not a solution to the differential equation.

STEP 23

After checking all options, we have determined that option C, y=xe2xy=x e^{2 x}, is the correct solution to the differential equation dydx=e2x+2y\frac{d y}{d x}=e^{2 x}+2 y.
The solution to the differential equation is y=xe2xy=x e^{2 x}.

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