PDE and FS Basics

Which of the following is an example for first order linear partial differential equation?

• Lagrange’s Partial Differential Equation

• Clairaut’s Partial Differential Equation

What is the nature of Lagrange’s linear partial differential equation?

• First-order, Second-degree

• Second-order, First-degree

• First-order, Third-degree

• First-order, First-degree

Find the general solution of the linear partial differential equation, yzp+zxq=xy.

• $\phi\left(x^2-y^2,y^2-z^2\right)=0$

• $\phi\left(x^2-z^2,\ y^2-z^2\right)=0$

• $\phi\left(x^2-y^2,-z^2\right)=0$

• $\phi\left(x^2-y^2,y^2-x^2\right)=0$

Which of the following equations represents Clairaut’s partial differential equation?

• z=px+f(p,q)

• z=f(p,q)

• z=p+q+f(p,q)

• z=px+qy+f(p,q)

The solution of Lagrange's partial differential equation xp+yq=z is

• $f\left(\frac{x}{y},\frac{y}{z}\right)=0$

• $f\left(\frac{x}{y},\frac{x}{z}\right)=0$

• $f\left(\frac{y}{x},\frac{y}{z}\right)=0$

• $f\left(\frac{y}{x},\frac{x}{z}\right)=0$

The solution of non-linear partial differential equation  

 $\sqrt{p}+\sqrt{q}=1$  is

•  $z=ax+\left(1-\sqrt{a}\right)^2\ y+c$  

•  $z=ax+ay+b$  

•  $z=ax+2ay+c$  

•  $z=\sqrt{a}x+ay+c$  

The solution of non-linear partial differential equation  

 $\sqrt{p}+\sqrt{q}=x$  is

•  $z=\left(x-a\right)^3+a^2y+b$  

•  $z=\frac{\left(x-a\right)^3}{3}+a^2y+b$  

•  $z=\left(x-a\right)^3+b$  

•  $z=a^2x+\left(y-a\right)^3+c$  

The solution of non-linear partial differential equation  

 $p\left(1+q\right)=qz$  is

•  $\log\left(az-1\right)=x+a\ y+c$  

•  $\log z=x+a\ y+c$  

•  $az-1=e^{x+a\ y+c}$  

The solution of non-linear partial differential equation  

 $q-p+x-y=0$  is

•  $z=\frac{\left(x+a\right)^2}{2}+\frac{\left(y+a\right)^2}{2}+c$  

•  $z=\left(x+a\right)^2+\left(y+a\right)^2+c$  

•  $z=\frac{\left(x+a\right)^2}{2}+\frac{\left(y-a\right)^2}{2}+c$  

•  $z=\frac{\left(x-a\right)^2}{2}+\frac{\left(y+a\right)^2}{2}+c$