Separable differential equations worksheet

When everything in x can be got to one side and everything in y to the other, both sides can be integrated on their own. The starting value is what turns the family of answers into a single one.

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Sigma Prep · sigmaprep.io/worksheets Name Separable Differential Equations Date Period Solve each equation. Solve $\frac{d y}{d x} = 4 x y$ with $y (0) = 6$ Solve $\frac{d y}{d x} = 3 x y$ with $y (0) = 8$ Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 4$ Solve $\frac{d y}{d x} = - 2 x y$ with $y (0) = 4$ Solve $\frac{d y}{d x} = - x y$ with $y (0) = 2$ Solve $\frac{d y}{d x} = - 2 x y$ with $y (0) = 3$ Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 5$ Solve $\frac{d y}{d x} = 3 x y$ with $y (0) = 3$ Solve $\frac{d y}{d x} = 2 x y$ with $y (0) = 7$ Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 7$ Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 2$ Solve $\frac{d y}{d x} = - x y$ with $y (0) = 5$ Solve $\frac{d y}{d x} = - 3 x y$ with $y (0) = 6$ Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 8$ Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 6$ Solve $\frac{d y}{d x} = 2 x y$ with $y (0) = 1$ Solve $\frac{d y}{d x} = x y$ with $y (0) = 8$ Solve $\frac{d y}{d x} = 4 x y$ with $y (0) = 5$ Solve $\frac{d y}{d x} = - 4 x y$ with $y (0) = 3$ Solve $\frac{d y}{d x} = - x y$ with $y (0) = 3$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Separable Differential Equations

Date Period

Solve each equation.

Solve $\frac{d y}{d x} = 4 x y$ with $y (0) = 6$
Solve $\frac{d y}{d x} = 3 x y$ with $y (0) = 8$
Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 4$
Solve $\frac{d y}{d x} = - 2 x y$ with $y (0) = 4$
Solve $\frac{d y}{d x} = - x y$ with $y (0) = 2$
Solve $\frac{d y}{d x} = - 2 x y$ with $y (0) = 3$
Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 5$
Solve $\frac{d y}{d x} = 3 x y$ with $y (0) = 3$
Solve $\frac{d y}{d x} = 2 x y$ with $y (0) = 7$
Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 7$
Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 2$
Solve $\frac{d y}{d x} = - x y$ with $y (0) = 5$
Solve $\frac{d y}{d x} = - 3 x y$ with $y (0) = 6$
Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 8$
Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 6$
Solve $\frac{d y}{d x} = 2 x y$ with $y (0) = 1$
Solve $\frac{d y}{d x} = x y$ with $y (0) = 8$
Solve $\frac{d y}{d x} = 4 x y$ with $y (0) = 5$
Solve $\frac{d y}{d x} = - 4 x y$ with $y (0) = 3$
Solve $\frac{d y}{d x} = - x y$ with $y (0) = 3$



Sigma Prep · sigmaprep.io/worksheets Name Separable Differential EquationsAnswer keyVersion 1 Date Period Solve each equation. Solve $\frac{d y}{d x} = 4 x y$ with $y (0) = 6$ $y = 6 e^{2 x^{2}}$ Solve $\frac{d y}{d x} = 3 x y$ with $y (0) = 8$ $y = 8 e^{\frac{3 x^{2}}{2}}$ Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 4$ $y = \sqrt{x^{2} + 16}$ Solve $\frac{d y}{d x} = - 2 x y$ with $y (0) = 4$ $y = 4 e^{- x^{2}}$ Solve $\frac{d y}{d x} = - x y$ with $y (0) = 2$ $y = 2 e^{- \frac{x^{2}}{2}}$ Solve $\frac{d y}{d x} = - 2 x y$ with $y (0) = 3$ $y = 3 e^{- x^{2}}$ Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 5$ $y = \sqrt{x^{2} + 25}$ Solve $\frac{d y}{d x} = 3 x y$ with $y (0) = 3$ $y = 3 e^{\frac{3 x^{2}}{2}}$ Solve $\frac{d y}{d x} = 2 x y$ with $y (0) = 7$ $y = 7 e^{x^{2}}$ Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 7$ $y = \sqrt{x^{2} + 49}$ Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 2$ $y = \sqrt{x^{2} + 4}$ Solve $\frac{d y}{d x} = - x y$ with $y (0) = 5$ $y = 5 e^{- \frac{x^{2}}{2}}$ Solve $\frac{d y}{d x} = - 3 x y$ with $y (0) = 6$ $y = 6 e^{- \frac{3 x^{2}}{2}}$ Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 8$ $y = \sqrt{x^{2} + 64}$ Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 6$ $y = \sqrt{x^{2} + 36}$ Solve $\frac{d y}{d x} = 2 x y$ with $y (0) = 1$ $y = e^{x^{2}}$ Solve $\frac{d y}{d x} = x y$ with $y (0) = 8$ $y = 8 e^{\frac{x^{2}}{2}}$ Solve $\frac{d y}{d x} = 4 x y$ with $y (0) = 5$ $y = 5 e^{2 x^{2}}$ Solve $\frac{d y}{d x} = - 4 x y$ with $y (0) = 3$ $y = 3 e^{- 2 x^{2}}$ Solve $\frac{d y}{d x} = - x y$ with $y (0) = 3$ $y = 3 e^{- \frac{x^{2}}{2}}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Separable Differential EquationsAnswer keyVersion 1

Date Period

Solve each equation.

Solve $\frac{d y}{d x} = 4 x y$ with $y (0) = 6$ $y = 6 e^{2 x^{2}}$
Solve $\frac{d y}{d x} = 3 x y$ with $y (0) = 8$ $y = 8 e^{\frac{3 x^{2}}{2}}$
Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 4$ $y = \sqrt{x^{2} + 16}$
Solve $\frac{d y}{d x} = - 2 x y$ with $y (0) = 4$ $y = 4 e^{- x^{2}}$
Solve $\frac{d y}{d x} = - x y$ with $y (0) = 2$ $y = 2 e^{- \frac{x^{2}}{2}}$
Solve $\frac{d y}{d x} = - 2 x y$ with $y (0) = 3$ $y = 3 e^{- x^{2}}$
Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 5$ $y = \sqrt{x^{2} + 25}$
Solve $\frac{d y}{d x} = 3 x y$ with $y (0) = 3$ $y = 3 e^{\frac{3 x^{2}}{2}}$
Solve $\frac{d y}{d x} = 2 x y$ with $y (0) = 7$ $y = 7 e^{x^{2}}$
Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 7$ $y = \sqrt{x^{2} + 49}$
Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 2$ $y = \sqrt{x^{2} + 4}$
Solve $\frac{d y}{d x} = - x y$ with $y (0) = 5$ $y = 5 e^{- \frac{x^{2}}{2}}$
Solve $\frac{d y}{d x} = - 3 x y$ with $y (0) = 6$ $y = 6 e^{- \frac{3 x^{2}}{2}}$
Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 8$ $y = \sqrt{x^{2} + 64}$
Solve $\frac{d y}{d x} = \frac{x}{y}$ with $y (0) = 6$ $y = \sqrt{x^{2} + 36}$
Solve $\frac{d y}{d x} = 2 x y$ with $y (0) = 1$ $y = e^{x^{2}}$
Solve $\frac{d y}{d x} = x y$ with $y (0) = 8$ $y = 8 e^{\frac{x^{2}}{2}}$
Solve $\frac{d y}{d x} = 4 x y$ with $y (0) = 5$ $y = 5 e^{2 x^{2}}$
Solve $\frac{d y}{d x} = - 4 x y$ with $y (0) = 3$ $y = 3 e^{- 2 x^{2}}$
Solve $\frac{d y}{d x} = - x y$ with $y (0) = 3$ $y = 3 e^{- \frac{x^{2}}{2}}$

How to do these

Solve $\frac{d y}{d x} = 2 y$ with $y (0) = 5$

Split it: dy over y equals 2 dx.
Integrate both sides: ln y equals 2x plus a constant.
So y is C e to the 2x, and y of 0 being 5 makes C equal 5.

The answer is $y = 5 e^{2 x}$ .

Where students go wrong

Adding the constant at the end instead of when the integrating happens. Once both sides are exponentiated the constant is a multiplier, not something added on, and putting it in late gives the wrong family of curves.

Also called separation of variables, initial value problem, solving dy/dx or first order differential equations.

Questions about these worksheets

Yes. Every separable differential equations sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the separable differential equations questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Separable differential equations worksheet

Select your difficulty

Separable Differential Equations

Separable Differential EquationsAnswer keyVersion 1

How to do these

Where students go wrong

Questions about these worksheets

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