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Separable equations

Splitting an equation into an x part and a y part, integrating each side in its own variable, and recovering solutions lost when dividing through.

Notes on Diffy Qs: Differential Equations for Engineers, Separable equations

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7 exercises

Exercise 101

Solve y′=2xyy'=2xy.

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Exercise 102

Solve x′=3xt2−3t2x'=3xt^2-3t^2, x(0)=2x(0)=2.

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Exercise 103

Find an implicit solution for x′=13x2+1x'=\dfrac{1}{3x^2+1}, x(0)=1x(0)=1.

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Exercise 104

Find an explicit solution to xy′=y2x y' = y^2, y(1)=1y(1) = 1.

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Exercise 105

Find an implicit solution to y′=sin⁡(x)cos⁡(y)y' = \dfrac{\sin(x)}{\cos(y)}.

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Exercise 106

Take an example in the source text with the same numbers: 89 degrees at t=0t=0, 85 degrees at t=1t=1, and ambient temperature of 22 degrees. Suppose these temperatures were measured with a precision of ±0.5\pm 0.5 degrees. Given this imprecision, the time it takes the coffee to cool to (exactly) 60 degrees is also only known in a certain range. Find this range. Hint: Think about what kind of error makes the cooli…

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Exercise 107

A population xx of rabbits on an island is modeled by x′=x−(1 ⁣/ ⁣1000)x2x' = x- \bigl(\nicefrac{1}{1000} \bigr) x^2, where the independent variable tt is time in months. At time t=0t=0, there are 40 rabbits on the island.

  1. Find the solution to the equation with the initial condition.
  2. How many rabbits are on the island in 1 month, 5 months, 10 months, 15 months (round to the nearest integer)?
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Adapted from Notes on Diffy Qs © 2008–2026 Jiří Lebl. Licensed under CC BY-SA 4.0. Source: Notes on Diffy Qs: Differential Equations for Engineers. Imported at revision 658bcae.