cheatmenot said:(a) all circles tangent to x-axis?? with answer on the book [1+(y^{2})^2]^3=[yy''+1+(y')^2]^2
(b) all circles with fixed radius r and tangent to x-axis??
Thanks a bunch!
mathmari said:(a) The equation of a circle with center at $(x_0,y_0)$ and radius $r$ is: $$(x-x_0)^2+(y-y_0)^2=r^2$$
Since the circles should tangent to the $x-$axis, $y_0= \pm r$.
Therefore,
$$(x-x_0)^2+(y \mp r)^2=r^2 $$
$x_0$ and $r$ are arbitrary constants and must not appear in ODE which generates the family of the circles. Since there are two arbitrary constants to eliminate, the ODE will be second order.
Differentiating both sides with respect to $x$ we have the following:
$$2(x-x_0)+2(y \mp r)y'=0 $$
Differentiating again with respect to $x$ we have the following:
$$2+2(y')^2+2(y \mp r)y''=0 $$
Now you have to use these equations to get rid of $x_0$ and $r$, or to get rid of $(x-x_0)$ and $(y \mp r)$ by expressing them in terms of $y$ and its derivatives.
(b)The equation of a circle with center at $(x_0,y_0)$ and fixed radius $r$ is: $$(x-x_0)^2+(y-y_0)^2=r^2$$
Since the circles should tangent to the $x-$axis, $y_0= \pm r$.
Therefore,
$$(x-x_0)^2+(y \mp r)^2=r^2 $$
$x_0$ is an arbitrary constant and must not appear in ODE which generates the family of the circles. Since there is only one arbitrary constant to eliminate, the ODE will be 1st order.
Differentiating both sides with respect to $x$ we have the following:
$$2(x-x_0)+2(y \mp r)y'=0 $$
Now you have to use these equations to get rid of $x_0$, or to get rid of $(x-x_0)$ by expressing this in terms of $r$ and of $y$ and its derivatives.
The process for obtaining differential equations with solution involves first identifying the variables and parameters involved in the system. Then, using any relevant laws or principles, such as Newton's Laws or the Law of Conservation of Energy, the equations can be set up. Finally, the equations can be solved using various methods, such as separation of variables or using a computer program.
There are several types of differential equations, including ordinary differential equations, which involve a single independent variable, and partial differential equations, which involve multiple independent variables. Other types include linear and nonlinear differential equations, as well as first-order and higher-order equations.
To check if a solution to a differential equation is correct, you can substitute the solution back into the original equation and see if it satisfies the equation. Additionally, solutions can be graphed or compared to known solutions to check for accuracy.
Yes, differential equations can be solved analytically, using methods such as separation of variables or integrating factors, or numerically, using computational methods such as Euler's method or Runge-Kutta methods. The method used depends on the complexity of the equation and the availability of initial conditions.
Differential equations have many practical applications in fields such as physics, engineering, and economics. They are used to model physical systems, such as the motion of objects or the flow of fluids, and to make predictions about their behavior. They are also used in the analysis of economic systems and population dynamics.