solve the differential equation.

$x + 6)y = 2y - 1\

\(y(x) =$

use your calculator to draw a family of solutions.

Question

solve the differential equation.

$x + 6)y = 2y - 1\

\(y(x) =$

use your calculator to draw a family of solutions.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Separable Differential Equations" ], "new_concepts": [], "current_concepts": [ "Separable Differential Equations", "First-Order Linear Differential Equations" ] } ## Separate variables and integrate $$ \begin{aligned} (x + 6)\frac{dy}{dx} &= 2y - 1 \ \int \frac{1}{2y - 1} \, dy &= \int \frac{1}{x + 6} \, dx \ \frac{1}{2} \ln|2y - 1| &= \ln|x + 6| + C_0 \end{aligned} $$ ## Solve for y $$ \begin{aligned} \ln|2y - 1| &= 2\ln|x + 6| + 2C_0 \ |2y - 1| &= e^{2C_0} (x + 6)^2 \ 2y - 1 &= C(x + 6)^2 \ y &= \frac{1}{2} + C(x + 6)^2 \end{aligned} $$ Solve the differential equation. $$(x + 6)y' = 2y - 1$$ $y(x) =$ $\frac{1}{2} + C(x + 6)^2$ { "elements": [ { "type": "functiongraph", "params": [{"js": "0.5 + 0.2*Math.pow(x+6, 2)", "latex": "0.5 + 0.2(x+6)^2"}, -10, 10], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2 } }, { "type": "functiongraph", "params": [{"js": "0.5 + 0.5*Math.pow(x+6, 2)", "latex": "0.5 + 0.5(x+6)^2"}, -10, 10], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "functiongraph", "params": [{"js": "0.5 - 0.2*Math.pow(x+6, 2)", "latex": "0.5 - 0.2(x+6)^2"}, -10, 10], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2 } }, { "type": "functiongraph", "params": [{"js": "0.5 - 0.5*Math.pow(x+6, 2)", "latex": "0.5 - 0.5(x+6)^2"}, -10, 10], "properties": { "strokeColor": "#F28A55", "strokeWidth": 2 } }, { "type": "point", "params": [[-6, 0.5]], "properties": { "name": "(-6, 1/2)", "size": 4, "color": "#583C87", "withLabel": true } } ], "timestamps": [0.1] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Separable Differential Equations" ] }

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Question

Explanation:

Separate variables and integrate

Solve for y

Answer: