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{\bf ASSIGNED EXERCiSES4: \\PDE Set  2\\} 
{\it February, 2001}
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\noindent
{\bf  PDE Problems}\\
Copson: \\
See also Ch.4 of the lecture notes.

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\noindent
{\bf 1}.\, Find the types of the following differential equations, and reduce
them to normal form:

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\begin{eqnarray}
{i}\,\,\,&& y^2 r-t=0,\\
(ii)\,\,&& y^2 r+2 y s+t=p,\\
(iii)\, &&y^2 r+t=0,\\
(iv)\,&& r-2s+3t-q-u=0,\\
(v)\,&& (1+x^2)^2 r-t=0,\\
(vi)\,&&x^2 r-2xys+y^2 +y^2t=0\\
(vii)\,&& x^2 r+2xys+y^2 t=0.
\end{eqnarray}

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\noindent
{\bf 2}.\, Reduce the folloeing equations to normal form:

\begin{eqnarray}
(i)\,\,&& u_{xx}+2u_{yy}++3u_{zz}+4u_{tt}+2u_{xy}+2u_{xz}+2u_{xt}+ \nonumber\\
&+&4u_{yz}+4u_{yt}+6u_{xt}=0,\\
(ii)\,\,&&u_{xx}+u_{zz}+2u_{xy}+2u_{xz}+2u_{xt}+2u_{zt}=0,\\
(iii)\,&&u_{xx}+u_{xy}+u_{xz}+u_{xt}=0.
\end{eqnarray}

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\noindent
{\bf 3}.\, Prove that $q^2 r-2pqs+p^2 t=0$ has an intemediate integral
 $p=q f(u)$, where $f$ is an arbitrary function. Hence show that $y+x f(u)=
g(u)$,  where $g$ is another arbitrary function


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\noindent
{\bf 4}.\, Transform the equation $x^2 u_{xx}-y^2 u_{yy}=0$ into one with characteristic
variables. Hence show that $u=f(xy)+x g({y \over x})$ , where $f$ and $g$
are arbitrary functions.

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\noindent
{\bf 5}.\, Find the characteristics of $r-yt=0$. reduce this equation to normal
form when $y>0$.

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\noindent
{\bf 6}.\, Find the characteristics of $(1+x^2)r-(1+y^2)t=0$, and reduce this
equations to its normal form.

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\noindent
{\bf 7}. Prove that $x^2 r+2xy s+y^2 t=0$ has an intermediate integral
$$px+qy-u+f({y \over x})=0$$
where $f$ is an arbitrary function. Hence show that
$$u=f({y \over x})+x g({y \over x})$$
where $g$ is an arbitrary function.

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\noindent
{\bf 8}.\, Show that the equation $qr+(uq-p)a=upt$ has an intermediate integral
$p+qu=f(u)$ , where $f$ is an arbitarary function. Hence solve the equation.

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