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General Differential Equations worksheets

Find exact general solutions of first-order differential equations whose right side is polynomial, rational, or trigonometric.

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Find the general solution on an interval of the original domain. Include one arbitrary constant C.

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Integrate the right side

Each equation has a right side depending only on x. Apply the matching polynomial, reciprocal, or trigonometric antiderivative rule and add one arbitrary constant.

Learning objective

What this activity builds

Integrate a right side depending only on x, add one arbitrary constant, and preserve the original connected-domain restriction.

Worked example

Solve dy/dx = 3x^2.

y = x^3 + C.

Integrate both sides with respect to x. Differentiating x^3 + C returns 3x^2.

Teaching notes

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  • Treat y or f as the unknown function and integrate the entire right side.
  • Keep excluded points and trigonometric poles in the original domain.
  • The constant may differ on separate connected domain intervals.

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Choose 8, 10, or 12 equations with polynomial, rational, or trigonometric right sides and prime, Leibniz, or mixed notation.

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Math SheetNameScoreTeacherDateGeneral Differential EquationsFind the general solution on an interval of the original domain. Include one arbitrary constant C.The arbitrary constant C may differ between connected intervals separated by an excluded point or trigonometric pole.1.dydx=-1General solution:2.dydx=-8xGeneral solution:3.dydx=12·cos2+2·xGeneral solution:4.dydx=-8·x3+9·x2General solution:5.dydx=-5x5General solution:6.dydx=12·sin1-2·xGeneral solution:7.dydx=-23-1·x-3General solution:8.dydx=-1·sec3·x2General solution:9.dydx=-2·x3+6·x2+4General solution:10.dydx=-1·csc3+3·x2General solution:math-sheet.comPage 1 of 1

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