Question #105517
Given that z = ax +sqrt (a^2 − 4y) + c is the complete integral of the PDE, p^2 − q^2 =4
determine its general integral.
1
Expert's answer
2020-05-21T12:21:57-0400

Correction.

To understand the recall some theory.

Equation of the form f(p,q)=0f(p,q)=0 i.e. the equation contains pp and qq only.

Suppose that z(x,y)=ax+by+cz(x,y)=ax+by+c is solution of equation, then



p=zx=aandq=zy=bp=\frac{\partial z}{\partial x}=a\,\,\,\text{and}\,\,\,q=\frac{\partial z}{\partial y}=b

substitute the above in equation f(p,q)=0f(p,q)=0 , we get f(a,b)=0f(a,b)=0 on solving this we can get b=ϕ(a)b=\phi(a) , where ϕ\phi is known function. Using b=ϕ(a)b=\phi(a) , the complete solution of the given partial equation is



z=ax+ϕ(a)y+cz=ax+\phi(a)y+c

In our case,



p2q2=4p2q24f(p,q)=0p^2-q^2=4\longrightarrow\underbrace{ p^2-q^2-4}_{f(p,q)}=0

Then, suppose



z=ax+by+cp=zx=aandq=zy=bp2q24=0a2b24=0b=±a24,a2z=ax+by+c\\[0.3cm] p=\frac{\partial z}{\partial x}=a\,\,\,\text{and}\,\,\,q=\frac{\partial z}{\partial y}=b\\[0.3cm] p^2-q^2-4=0\longrightarrow a^2-b^2-4=0\longrightarrow\\[0.3cm] \boxed{b=\pm\sqrt{a^2-4},\,\,\,\forall |a|\ge2}

Conclusion,



z=ax±a24y+cis the general integral\boxed{z=ax\pm\sqrt{a^2-4}\cdot y+c\,\,\,-\,\,\text{is the general integral}}



and



z=ax+a24y+cz=ax+\sqrt{a^2-4}\cdot y+c

is the complete integral as a special case.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS