Differential Equations : Solution Curves

In summary, the conversation discusses solving the differential equation (y')^2= 4y and verifying the general and singular solution curves. The problem also involves determining the points (a,b) in the plane for which the initial value problem (y')^2= 4y, y(a)= b has no solution, infinitely many solutions, or only finitely many solutions on a neighborhood of x=a. The general solution is y(x) = (x-c)^2 and the singular solution is y(x)=0. The solution for part (a) is b < 0. For parts (b) and (c), the function g(x) is considered, with g(x)=0 for x < c and g(x
  • #1
mehtamonica
26
0
I have to solve the differential equation (y')^2= 4y to verify the general solution curves and singular solution curves.
Determine the points (a,b) in the plane for which the initial value problem (y')^2= 4y, y(a)= b has
(a) no solution ,
(b) infinitely many solutions (that are defined for all values of x )
(c) on some neighborhood of the point x=a , only finitely many solutions.

general solution that i am getting is y (x) = (x-c)^2 and singular solution is y(x)=0.

I am able to get part (a), as if b < 0, the problem has no solution.

Please help me figure out (b) and (c) .
 
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  • #2
Think about a function g(x) defined piecewise with g(x) = 0 for x < c and g(x) = (x-c)2 if x ≥ c.
 
  • #3
For (c) consider the situation when b= 0.
 

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of calculus to determine the rate of change of a variable over time.

2. How are differential equations solved?

There is no one specific method for solving differential equations, as it depends on the type and complexity of the equation. Some common methods include separation of variables, substitution, and using integrating factors.

3. What are solution curves?

Solution curves are graphical representations of the solutions to a differential equation. They show how the dependent variable changes in relation to the independent variable.

4. How do you determine the direction of a solution curve?

The direction of a solution curve can be determined by looking at the slope of the curve at a given point. If the slope is positive, the curve is increasing, and if the slope is negative, the curve is decreasing.

5. Can you provide an example of a real-world application of differential equations?

Differential equations are used in many fields, including physics, engineering, and biology. One example of a real-world application is using differential equations to model population growth or decay in a biological system.

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