Sketching Solutions on Slope Fields

Calculus 2 · Axiom Academy

EXAMPLE Sketching Solutions on Slope Fields Read a first-order slope field and trace a solution curve through a point — then watch every curve funnel toward one line. Consider the differential equation . Using its slope field, sketch the solution curve passing through the initial point (0, 1) , and describe how solution curves behave as x grows large. Nice work. You sketched a solution straight from a slope field — no formula required — and read off its long-run behavior. A slope field draws the equation: at every point (x, y) , a short segment of slope shows which way a solution must go there. Follow the flow: a solution curve through a point is drawn by staying tangent to the nearby segments — here it leaves (0, 1) with slope -1 , dips, then turns upward. Flat-slope line vs. solution: on the line y = x , but that line is not itself a solution; the line y = x - 1 is a solution. Asymptotic behavior: every solution is y = x - 1 + Ce^ -x , and since , all curves approach y = x - 1 as regardless of the starting point. Slope fields let you read the qualitative story of a differential equation — where solutions rise, fall, and settle — even when you never solve it on paper.

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