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Solving Separable Equations worksheets

Solve genuinely y-dependent first-order separable equations with exact implicit solutions, connected-domain restrictions, and restored equilibrium solutions.

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Separate variables, integrate both sides, and give an implicit general solution with one C and every equilibrium solution.

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Separate before integrating

These equations use both multiplication and division by y-dependent power, exponential, and squared trigonometric factors. Move the y factor beside dy and the x factor beside dx, then integrate each side exactly.

Learning objective

What this activity builds

Separate variables, integrate exact polynomial, exponential, and trigonometric factors, and report both nonconstant branches and every lost equilibrium.

Worked example

Solve dy/dx = x y^2.

-1/y = x^2/2 + C, together with y = 0.

Divide by y^2 for nonconstant branches, integrate y^(-2) dy = x dx, then restore y = 0 because division removed it.

Teaching notes

Make the practice count

  • State restrictions from the original equation separately from restrictions introduced by division.
  • Keep the implicit form when it preserves all real branches clearly.
  • Check the result by differentiating the implicit equality and substituting the original ODE.

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Choose 12 or 24 genuinely y-dependent equations from polynomial or exponential and trigonometric groups. Every key includes domains, equilibrium solutions, and a differentiated implicit check.

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Math SheetNameScoreTeacherDateSolving Separable EquationsSeparate variables, integrate both sides, and give an implicit general solution with one C and every equilibrium solution.Interpret nonconstant branches on connected intervals in the original domain and where division by the y-factor is valid.1.dydx=-1·cos3·x+1y2Implicit solution and equilibria:2.dydx=-9·x·yImplicit solution and equilibria:3.dydx=-7·e-2·x·y2Implicit solution and equilibria:4.dydx=-5·x3yImplicit solution and equilibria:5.dydx=-3·sinx-3·y3Implicit solution and equilibria:6.dydx=-12·x·y2Implicit solution and equilibria:7.dydx=32·x·e3·yImplicit solution and equilibria:8.dydx=5·x3y2Implicit solution and equilibria:9.dydx=-23·sec-1·x2·y3Implicit solution and equilibria:10.dydx=5·x3·y3Implicit solution and equilibria:11.dydx=x4·sin3·y2Implicit solution and equilibria:12.dydx=-2·xy3Implicit solution and equilibria:math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateSolving Separable EquationsSeparate variables, integrate both sides, and give an implicit general solution with one C and every equilibrium solution.Interpret nonconstant branches on connected intervals in the original domain and where division by the y-factor is valid.13.dydx=32·csc-1·x2·yImplicit solution and equilibria:14.dydx=6·x0·yImplicit solution and equilibria:15.dydx=12·exy2Implicit solution and equilibria:16.dydx=3·xyImplicit solution and equilibria:17.dydx=23·sec-3·x-1·tan-3·x-1y3Implicit solution and equilibria:18.dydx=7·x3·y2Implicit solution and equilibria:19.dydx=-4·x0·e-1·yImplicit solution and equilibria:20.dydx=3·xy2Implicit solution and equilibria:21.dydx=csc-x+1·cot-x+1·y2Implicit solution and equilibria:22.dydx=4·x2·y3Implicit solution and equilibria:23.dydx=-3·x3·cos2·y2Implicit solution and equilibria:24.dydx=-4·x0y3Implicit solution and equilibria:math-sheet.comPage 2 of 2

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