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Differential Equations

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Reducing Other Types of Equations to ODEs

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The following equations are not ordinary differential equations. However, some of them can be regarded as ordinary differential equations under some circumstances. Regarding this fact, select the one that is wrong.

A

Consider the partial differential equation (PDE)

$$u_t=u(1-u)+2x$$
where $u(t,x)$ is an unknown function. If we regard $x$ as a parameter, then $u$ is a function of $t$ only and we then have an ordinary differential equation.

B

The equation

$$y=\int_0^xy(s)ds+2x$$
is an integral equation. It is equivalent to the ordinary differential equation $y'=y+2$. Since $y(0)$ is unknown, the ODE has many solutions and hence the integral equation also has many solutions.

C

Consider $u=u(t,x)$ where $t$ is time and $x$ is position. The equation,

$$u_t-u_{xx}+u=f(x)$$
describes a physical process where diffusion and reaction are present. Its steady solution can be obtained by solving
$$-u_{xx}+u=f(x)$$
which is an ordinary differential equation.

D

The KDV equation

$$u_t+u_{xxx}+6uu_x=0$$

is a nonlinear partial differential equation about $u=u(x,t)$. It is known that this equation has a family of solutions of the form

$$u(x,t)=y(x-ct-a)$$

Then, the solutions with $c=2$ can be found by solving an ODE for $y$:
$$-2y'+y'''+6yy'=0$$