Solution of Two dimensional Laplace equation.

Q. The general integral of the partial differential equation $y^{2}p-xyq=x(z-2y)$ is

1. $\varphi (x^2+y^2,y^2-yz)=0$
2. $\varphi (x^2-y^2,y^2+yz)=0$
3. $\varphi (xy,yz)=0$
4. $\varphi (x+y,Inx-z)=0$

Q. The partial differential equation

$\frac{{\partial }^{2}u}{\partial t^2}-C^2(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2})=0$ ; where c $\ne$ 0 is known as

1. wave equation
2. laplace equation
3. heat equation
4. poission's equation

Q.

Which expression is equivalent to the given expression?

Assume the denominator does not equal zero.

$\frac{14x^4{y}^6}{7x^8{y}^2}$

1. $\frac{2y^4}{x^4}$

2. $\frac{2y^3}{x^2}$

3. $7x^4{y}^4$

4. $2x^2{y}^3$

Q. The solution at $x=1,t=1$ of the partial differential equation $\frac{{\partial }^{2}u}{\partial x^2}=25\frac{{\partial }^{2}u}{\partial t^2}$ subject to initial conditions of $u\left(0\right)=3x$ and $\frac{\partial u}{\partial t}\left(0\right)=3$ is

1. $1$
2. $2$
3. $4$
4. $6$

Q. The complete integral of
$(z-px-qy{)}^3=pq+2(p^2+q{)}^2$ is

1. $z=ax+by+\sqrt[3]{3pq+2(p^2+q{)}^2}$
2. $z=ax+by+\sqrt[3]{3ab+2(a^2+b{)}^2}$
3. $z=ax+by+\sqrt[3]{3ab}+\sqrt[3]{32(a^2+b{)}^2}$
4. $z=ax+by+c$