Hw1 diff eq - Homework One PDF

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Homework One
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21-260

Differential Equations

D. Handron

Written Homework: Week #1 September 4, 2020 1. Given the differential equation and function dy + 20y = 24, dt

φ(t) =

6 6 −20t − e 5 5

do the following: (a) Identify the dependent and independent viariable(s). (b) Classify the equation as linear/non-linear, and ODE/PDE. What is the order of the equation? (c) Show that y = φ(t) is a solution to the differential equation.

2. Given the initial value problem dy = 2xy 2 , dx

y(1) =

1 3

do the following: (a) Identify the dependent and independent variable(s). (b) Classify the equation as linear/non-linear, and ODE/PDE. What is the order of the equation? (c) Show that φ(x) =

1 4−x2

is a solution to the differential equation.

3. Find all solutions to the differential equation x

1 dy = (1 − y 2 ) 2 dx

4. Find a solution, φ(x), to the initial value problem y ′ = 2y 2 + xy 2 ,

y(0) = 1

Indicate the domain of definition for the solution you have found. Where does this solution attains its minimum value? (You should be able to do this without finding φ′ (x).)

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5. Consider the differential equation y − 4x dy = . dx x−y (a) Show that the equation can be written in the form (y/x) − 4 dy . = dx 1 − (y/x) this is an example of a homogeneous differential equation. (b) Introduce a new dependent variable v = v(x) satisfying v = y/x, or y = xv(x). Express dy/dx in terms of x, v, and dv/dx. (c) Rewrite the original differential equation in terms of v, x, and dv/dx by substituting y = xv (x), and the expression you found for dy/dx. Show that the result is a separable differential equation. (d) Solve the separable differential equation you found in part (5c). (e) Find the solution to the original differential equation by substituting y/x in place of v in the solution you found in part (5d).

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