The solutions of initial value problems for linear first order differential equations are defined on the same domain, no matter what are the values of the initial condition. This property is not true for solutions of nonlinear first order differential equations. In the problem below we solve nonlinear differential equation with two different initial conditions. We obtain two different solutions, called y₁ (t) and y₂ (t). These two solutions are defined on different domains. Therefore, this problem is an example of a nonlinear differential equation with solutions defined on domains that depend on the choice of initial conditions. Consider the differential equation for the function y y = 2y³. Find an implicit expression for all solutions y of the differential equation above, in the form (t, y) = c, where c collects all integration constants. (t, y) = Σ Note: Do not include the constnat c in your answer. (1a) Find the explicit expression for a solution y₁ (t) of differential equation above that satisfies the initial condition 3₁ (0) = 1. y₁ (t) = (1b) Find the largest value ₁ such that the solution ₁ is defined on [0,t₁). t₁ = Σ (2a) Find the explicit expression for a solution y₂ (t) of differential equation above that satisfies the initial condition 3/₂ (0) = 2. Y₂ (t) = (2b) Find the largest value to such that the solution y2 is defined on [0, t₂). t₂ = Σ
The solutions of initial value problems for linear first order differential equations are defined on the same domain, no matter what are the values of the initial condition. This property is not true for solutions of nonlinear first order differential equations. In the problem below we solve nonlinear differential equation with two different initial conditions. We obtain two different solutions, called y₁ (t) and y₂ (t). These two solutions are defined on different domains. Therefore, this problem is an example of a nonlinear differential equation with solutions defined on domains that depend on the choice of initial conditions. Consider the differential equation for the function y y = 2y³. Find an implicit expression for all solutions y of the differential equation above, in the form (t, y) = c, where c collects all integration constants. (t, y) = Σ Note: Do not include the constnat c in your answer. (1a) Find the explicit expression for a solution y₁ (t) of differential equation above that satisfies the initial condition 3₁ (0) = 1. y₁ (t) = (1b) Find the largest value ₁ such that the solution ₁ is defined on [0,t₁). t₁ = Σ (2a) Find the explicit expression for a solution y₂ (t) of differential equation above that satisfies the initial condition 3/₂ (0) = 2. Y₂ (t) = (2b) Find the largest value to such that the solution y2 is defined on [0, t₂). t₂ = Σ
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:The solutions of initial value problems for linear first order differential equations are defined on the same domain, no matter what are the values of the initial condition. This property is not true for solutions
of nonlinear first order differential equations.
In the problem below we solve a nonlinear differential equation with two different initial conditions. We obtain two different solutions, called y₁ (t) and y₂ (t). These two solutions are defined on different
domains.
Therefore, this problem is an example of a nonlinear differential equation with solutions defined on domains that depend on the choice of initial conditions.
Consider the differential equation for the function y
y' = 2y³.
Find an implicit expression for all solutions y of the differential equation above, in the form (t, y) = c, where c collects all integration constants.
(t, y) =
Σ
Note: Do not include the constnat c in your answer.
(1a) Find the explicit expression for a solution y₁ (t) of differential equation above that satisfies the initial condition
y₁ (0) = 1.
y₁ (t) =
(1b) Find the largest value ₁ such that the solution ₁ is defined on [0,t₁).
t₁ =
Σ
Σ
(2a) Find the explicit expression for a solution y₂ (t) of differential equation above that satisfies the initial condition
3/₂ (0) = 2.
Y₂ (t) =
Σ
(2b) Find the largest value to such that the solution y2 is defined on [0, t₂).
t₂ =
Σ
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