Math / Calculus

QuestionConsider the differential equation $\frac{d y}{d x}=\frac{5 x-1}{-6 y+4}$ , where $y \neq \frac{2}{3}$ . The slope field for the given differential equation is shown below. Sketch the solution curve that passes through the point $(2,-1)$ . undo last open circle erase all drawings
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Studdy Solution

STEP 1

1. The differential equation is separable, allowing us to separate variables for integration.
2. The solution curve must pass through the given point $(2, -1)$ .
3. The slope field provides visual guidance for sketching the solution curve.

STEP 2

1. Analyze the differential equation and separate variables.
2. Integrate both sides to find the general solution.
3. Apply the initial condition to find the particular solution.
4. Sketch the solution curve on the slope field.

STEP 3

Separate the variables in the differential equation $\frac{dy}{dx} = \frac{5x-1}{-6y+4}$ :
Rearrange to get:
$(-6y + 4) \, dy = (5x - 1) \, dx$

STEP 4

Integrate both sides to find the general solution:
Integrate the left side with respect to $y$ :
$\int (-6y + 4) \, dy$
Integrate the right side with respect to $x$ :
$\int (5x - 1) \, dx$

STEP 5

Calculate the integrals:
Left side:
$\int (-6y + 4) \, dy = -3y^2 + 4y + C_1$
Right side:
$\int (5x - 1) \, dx = \frac{5}{2}x^2 - x + C_2$
Combine the results:
$-3y^2 + 4y + C_1 = \frac{5}{2}x^2 - x + C_2$

STEP 6

Apply the initial condition $(2, -1)$ to find the particular solution:
Substitute $x = 2$ and $y = -1$ into the equation:
$-3(-1)^2 + 4(-1) + C_1 = \frac{5}{2}(2)^2 - 2 + C_2$
Simplify and solve for the constant:
$-3 - 4 + C_1 = 10 - 2 + C_2$ $C_1 - 7 = C_2 + 8$ $C_1 - C_2 = 15$

STEP 7

Sketch the solution curve on the slope field:
1. Start at the point $(2, -1)$ .
2. Follow the direction of the slope lines, ensuring the curve aligns with the slope field.
3. Extend the curve to the edges of the grid, maintaining the pattern indicated by the slope field.
The solution curve passing through $(2, -1)$ is sketched on the slope field, following the direction of the slope lines.

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Studdy Solution

STEP 1

1. The differential equation is separable, allowing us to separate variables for integration.
2. The solution curve must pass through the given point $(2, -1)$ .
3. The slope field provides visual guidance for sketching the solution curve.

STEP 2

1. Analyze the differential equation and separate variables.
2. Integrate both sides to find the general solution.
3. Apply the initial condition to find the particular solution.
4. Sketch the solution curve on the slope field.

STEP 3

Separate the variables in the differential equation $\frac{dy}{dx} = \frac{5x-1}{-6y+4}$ :
Rearrange to get:
$(-6y + 4) \, dy = (5x - 1) \, dx$

STEP 4

Integrate both sides to find the general solution:
Integrate the left side with respect to $y$ :
$\int (-6y + 4) \, dy$
Integrate the right side with respect to $x$ :
$\int (5x - 1) \, dx$

STEP 5

Calculate the integrals:
Left side:
$\int (-6y + 4) \, dy = -3y^2 + 4y + C_1$
Right side:
$\int (5x - 1) \, dx = \frac{5}{2}x^2 - x + C_2$
Combine the results:
$-3y^2 + 4y + C_1 = \frac{5}{2}x^2 - x + C_2$

STEP 6

Apply the initial condition $(2, -1)$ to find the particular solution:
Substitute $x = 2$ and $y = -1$ into the equation:
$-3(-1)^2 + 4(-1) + C_1 = \frac{5}{2}(2)^2 - 2 + C_2$
Simplify and solve for the constant:
$-3 - 4 + C_1 = 10 - 2 + C_2$ $C_1 - 7 = C_2 + 8$ $C_1 - C_2 = 15$

STEP 7

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Studdy Solution

STEP 1

1. The differential equation is separable, allowing us to separate variables for integration.2. The solution curve must pass through the given point (2,−1)(2, -1)(2,−1).3. The slope field provides visual guidance for sketching the solution curve.

STEP 2

1. Analyze the differential equation and separate variables.2. Integrate both sides to find the general solution.3. Apply the initial condition to find the particular solution.4. Sketch the solution curve on the slope field.

STEP 3

Separate the variables in the differential equation dydx=5x−1−6y+4\frac{dy}{dx} = \frac{5x-1}{-6y+4}dxdy​=−6y+45x−1​:Rearrange to get:(−6y+4) dy=(5x−1) dx (-6y + 4) \, dy = (5x - 1) \, dx (−6y+4)dy=(5x−1)dx

STEP 4

Integrate both sides to find the general solution:Integrate the left side with respect to y y y:∫(−6y+4) dy \int (-6y + 4) \, dy ∫(−6y+4)dyIntegrate the right side with respect to x x x:∫(5x−1) dx \int (5x - 1) \, dx ∫(5x−1)dx

STEP 5

STEP 6

STEP 7

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1. The differential equation is separable, allowing us to separate variables for integration.
2. The solution curve must pass through the given point $(2, -1)$ .
3. The slope field provides visual guidance for sketching the solution curve.

1. Analyze the differential equation and separate variables.
2. Integrate both sides to find the general solution.
3. Apply the initial condition to find the particular solution.
4. Sketch the solution curve on the slope field.

Separate the variables in the differential equation $\frac{dy}{dx} = \frac{5x-1}{-6y+4}$ :
Rearrange to get:
$(-6y + 4) \, dy = (5x - 1) \, dx$

Integrate both sides to find the general solution:
Integrate the left side with respect to $y$ :
$\int (-6y + 4) \, dy$
Integrate the right side with respect to $x$ :
$\int (5x - 1) \, dx$