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solve the given initial value problem and determine how the interval in which the solution exists depends on the initial value y0. y=2ty2,y(0)=y0

Short Answer

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Question: Consider the following Initial Value Problem (IVP) involving a first-order differential equation: y=2ty2, and the initial value is provided as y(0)=y0. Determine the solution to the IVP and the interval on which the solution exists based on the initial value y0. Answer: The solution to the IVP is given by y(t)=11y0t2. The interval of existence for the solution depends on the initial value y0. For y0>0, the interval of existence is (1y0,1y0), and for y0<0, the interval of existence is (,1|y0|)(1|y0|,). If y0=0, no interval of existence can be determined as the denominator will always be zero.

Step by step solution

01

Separate Variables

Begin by separating the variables, which will allow us to integrate both sides of the equation. We can rewrite the equation as: dyy2=2tdt
02

Integrate Both Sides

Now, we will integrate both sides of the equation with respect to their respective variables: dyy2=2tdt Upon integrating, we'll obtain: 1y=t2+C To find the constant of integration, C, we'll use the given initial condition, y(0)=y0.
03

Apply Initial Condition and Solve for C

Plug in the initial condition, y(0)=y0, into the equation: 1y0=(0)2+C From this, we can find the value of C: C=1y0
04

Replace C in the Solution and Rewrite the Solution

Now, we'll replace the 'C' in our solution with its value, and rewrite the solution as a function of 'y': 1y=t21y0 Rearrange to find the function y(t): y(t)=11y0t2
05

Determine the Interval of Existence

The solution for 'y(t)', y(t)=11y0t2, is valid as long as the denominator is non-zero. Therefore, we have: 1y0t20 Which yields: t21y0 In order for the solution to exist, the interval in which 't' can exist depends on the initial value y0: t2<1y0,y0>0 t2>1y0,y0<0 For y0>0, the interval of existence is (1y0,1y0), and for y0<0, the interval of existence is (,1|y0|)(1|y0|,). Note that if y0=0, no interval of existence can be determined because the denominator will always be zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations
Differential equations are mathematical equations that relate some function with its derivatives. They play a crucial role in engineering, physics, economics, and other sciences to model the behavior of complex systems. The given exercise involves an initial value problem for a differential equation, which is a special case where you know the value of the function or its derivatives at a certain point.

The equation in our problem, y\textprime=2ty2, is an ordinary differential equation (ODE) because it contains one independent variable, t, and it is first-order because it involves the first derivative y\textprime. Our goal is to find a solution y(t) that satisfies both the equation and the initial condition given by y(0)=y0.
Separation of Variables
Separation of variables is a method commonly used to solve straightforward first-order ODEs, such as the one presented in our exercise. This method involves rearranging the equation so that all terms involving the dependent variable y are on one side, and all terms involving the independent variable t are on the other.

In this case, we separated the variables by dividing by y2 and multiplying by dt, resulting in dy/y2=2tdt. This form allows us to integrate each side independently. This step is crucial for turning the differential equation into a solvable algebraic equation.
Integration of Differential Equations
After separating the variables in our ODE, the next step is to integrate both sides. Integration is the inverse operation of differentiation and helps us to find the original function given the derivative.

For the integration step in our problem, we calculate the integral of each side separately, giving 1/y=t2+C, where C is the constant of integration. This constant is determined using the initial condition, which is a distinctive value of the function at a particular point. Here, the initial condition y(0)=y0 helps us find the specific solution to the ODE that we're looking for.
Existence Interval
The existence interval of the solution to an initial value problem indicates where the solution is valid. For a solution to exist, certain conditions must be met. Specifically, the solution and its derivatives must be continuous and finite within the interval.

For the solution y(t)=1/(1y0t2), the denominator must not be zero. The conditions t2<1/y0 for y0>0 and t2>1/y0 for y0<0 define the intervals where the solution exists. The existence interval varies depending on the initial value y0, and it is essential to consider this when analyzing the solution's behavior at different values of t.

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Most popular questions from this chapter

transform the given initial value problem into an equivalent problem with the initial point at the origin. dy/dt=t2+y2,y(1)=2

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