Second order linear partial differential equations;

Q. The type of the partial differential equation $\frac{\partial f}{\partial t}=\frac{{\partial }^{2}f}{\partial x^2}$ is

1. Parabolic
2. Elliptic
3. Hyperbolic
4. Nonlinear

Q. Consider the following partial differential equation $u(x,y)$ with the constant $c>1$;
$\frac{\partial u}{\partial y}+c\frac{\partial u}{\partial x}=0$
Solution of this equation is

1. $u(x,y)=f(x+cy)$
2. $u(x,y)=f(x-cy)$
3. $u(x,y)=f(cx+y)$
4. $u(x,y)=f(cx-y)$

Q. For the second order linear ordinary differential equation.
$\frac{d^{2}y}{d{x}^2}+p\frac{dy}{dx}+qy=0$, where p and q real numbers.
The following function is a solution $y=e^{\lambda x}$
Which one of the following statements is Not TRUE?

1. $\lambda$ has two values : one complex and one real
2. ${\lambda }^2+p\lambda +q=0$
3. $\lambda$ has two real values
4. $\lambda$ has two complex values

Q. The type of partial differential equation,
$\frac{{\partial }^{2}P}{\partial x^2}+\frac{1}{2}\frac{{\partial }^{2}P}{\partial x\partial y}-\frac{5\partial P}{\partial x}+\frac{2\partial P}{\partial y}=0$ is

1. ellipltical
2. parabolic
3. hyperbolic
4. none