- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Slope fields and Euler’s method practice question 1
In each equation is an unknown function of . Which one of these differential equations is nonlinear?
Try to put each equation in the linear form. What is allowed to multiply or there, and what is not?
A linear equation can be written , where , , may depend on but not on . Look for a power of , inside another function, or times .
Theory UBC Math 100 §12.1Key results
Worked solution
Strategy
Try to write each equation as α dy/dt + βy + γ = 0. The coefficients may depend on t but not on y; an equation that cannot be put in this form is nonlinear.
Working
Constant coefficients: linear.
A coefficient that depends on t is allowed: linear.
The et contains no y: linear.
The y² term is not a multiple of y by anything free of y, so this one is nonlinear.
Answer
The answer is dy by dt equals 6 minus 2 y squared.
Check
6 − 2y² is zero at y = √3 and at y = −√3 but not at y = 0, which no linear expression a + by can do.
Where this goes wrong
Calling an equation nonlinear because t appears squared or in an exponential. Linearity is about y alone: t may appear in any form, but y and dy/dt must appear only to the first power.
Leans on
What a differential equation is, from Exponential growth and decay
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