Added April 13, 2019.
Problem Chapter 6.2.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y) + c*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},{\frac {za-cx}{a}} \right ) \]
Hand solution
Solve \begin {equation} aw_{x}+bw_{y}+cw_{z}=0\tag {1} \end {equation}
Using Lagrange-charpit
\[ \frac {dx}{a}=\frac {dy}{b}=\frac {dz}{c}=\frac {dw}{0}\]
From first two pair of equations, integrating gives \(\frac {b}{a}x-y=C_{1}\) and from \(\frac {dx}{a}=\frac {dz}{c}\) by Integrating gives \(\frac {c}{a}x-z=C_{2}\). Since \(dw=0\) then \(w=C_{3}\). Where \(C_{1},C_{2},C_{3}\) are constants. But \(C_{3}=F\left ( C_{1},C_{2}\right ) \) where \(F\) is arbitrary function. Hence
\[ u\left ( x,y,z\right ) =F\left ( \frac {b}{a}x-y,\frac {c}{a}x-z\right ) \]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 6.2.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a x w_y + b y w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*x*D[w[x, y,z], y] +b*y*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a x^2}{2},\frac {1}{3} a b x^3-b x y+z\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ a*x*diff(w(x,y,z),y) + b*y*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -{\frac {a{x}^{2}}{2}}+y,{\frac {x \left ( a{x}^{2}-3\,y \right ) b}{3}}+z \right ) \]
Hand solution
Solve \begin {equation} w_{x}+axw_{y}+byw_{z}=0\tag {1} \end {equation}
Using Lagrange-charpit
\[ dx=\frac {dy}{ax}=\frac {dz}{by}=\frac {dw}{0}\]
From first two pair of equations, integrating gives \begin {equation} a\frac {x^{2}}{2}-y=C_{1}\tag {1} \end {equation} And from \(dx=\frac {dz}{by}\) we obtain \[ bydx=dz \] But from (1) \(y=\frac {ax^{2}}{2}-C_{1}\) and the above becomes\[ b\left ( \frac {ax^{2}}{2}-C_{1}\right ) dx=dz \] Now we can integrate and the result is\[ b\left ( \frac {ax^{3}}{6}-C_{1}x\right ) -z=C_{2}\] Using (1) again the above becomes\begin {align*} b\left ( \frac {ax^{3}}{6}-\left ( a\frac {x^{2}}{2}-y\right ) x\right ) -z & =C_{2}\\ b\left ( \frac {ax^{3}}{6}-a\frac {x^{3}}{2}+yx\right ) -z & =C_{2}\\ -\frac {b}{3}ax^{3}+byx-z & =C_{2} \end {align*}
Since \(dw=0\) then \(w=C_{3}\). Where \(C_{1},C_{2},C_{3}\) are constants. But \(C_{3}=F\left ( C_{1},C_{2}\right ) \) where \(F\) is arbitrary function. Hence\begin {align*} u\left ( x,y,z\right ) & =F\left ( a\frac {x^{2}}{2}-y,-\frac {b}{3}ax^{3}+byx-z\right ) \\ & =F\left ( y-a\frac {x^{2}}{2},\frac {1}{3}abx^{3}-byx+z\right ) \end {align*}
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 6.2.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b y w_y + c z w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*y*D[w[x, y,z], y] +c*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y e^{-\frac {b x}{a}},z e^{-\frac {c x}{a}}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*y*diff(w(x,y,z),y) + c*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( y{{\rm e}^{-{\frac {xb}{a}}}},z{{\rm e}^{-{\frac {cx}{a}}}} \right ) \]
Hand solution
Solve \begin {equation} aw_{x}+byw_{y}+czw_{z}=0\tag {1} \end {equation}
Using Lagrange-charpit
\[ \frac {dx}{a}=\frac {dy}{by}=\frac {dz}{cz}=\frac {dw}{0}\]
From first two pair of equations, integrating gives \begin {align} abx+C_{1} & =\ln y\nonumber \\ y & =C_{1}e^{abx}\nonumber \\ C_{1} & =ye^{-abx}\tag {1} \end {align}
And from \(\frac {dx}{a}=\frac {dz}{cz}\) we obtain \begin {align*} \frac {c}{a}x+C_{2} & =\ln z\\ z & =C_{2}e^{\frac {c}{a}x}\\ C_{2} & =ze^{-\frac {c}{a}x} \end {align*}
Since \(dw=0\) then \(w=C_{3}\). Where \(C_{1},C_{2},C_{3}\) are constants. But \(C_{3}=F\left ( C_{1},C_{2}\right ) \) where \(F\) is arbitrary function. Hence\[ u\left ( x,y,z\right ) =F\left ( ye^{-abx},ze^{-\frac {c}{a}x}\right ) \]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 6.2.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a z w_y + b y w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*z*D[w[x, y,z], y] +b*y*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {b}},\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {a}}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ a*z*diff(w(x,y,z),y) + b*y*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {{z}^{2}a-b{y}^{2}}{a}},-{ \left ( -x\sqrt {ba}+\ln \left ( { \left ( aby+\sqrt {{a}^{2}{z}^{2}}\sqrt {ba} \right ) {\frac {1}{\sqrt {ba}}}} \right ) \right ) {\frac {1}{\sqrt {ba}}}} \right ) \]
Hand solution
Solve \[ w_{x}+azw_{y}+byw_{z}=0 \] Using Lagrange-charpit\[ dx=\frac {dy}{az}=\frac {dz}{by}=\frac {dw}{0}\] Starting with \(\frac {dy}{az}=\frac {dz}{by}\) or \(\frac {b}{a}ydy=zdz\) and integrating gives\begin {align} \frac {b}{a}\frac {y^{2}}{2} & =\frac {z^{2}}{2}+C_{1}\nonumber \\ \frac {b}{a}y^{2} & =z^{2}+C_{1}\nonumber \\ C_{1} & =\frac {b}{a}y^{2}-z^{2}\tag {1} \end {align}
Now we can either consider \(dx=\frac {dz}{by}\) or \(dx=\frac {dy}{az}\). Both will give valid solutions. Lets try both to see.
case \(dx=\frac {dz}{by}\)
From (1) solving for \(y\) in terms of \(z\) gives\[ \sqrt {\frac {a}{b}C_{1}+\frac {a}{b}z^{2}}=y \] Hence \(dx=\frac {dz}{by}\) becomes \(bdx=\frac {dz}{\sqrt {\frac {a}{b}C_{1}+\frac {a}{b}z^{2}}}\). Integrating gives\[ bx=\sqrt {\frac {b}{a}}\ln \left ( z+\sqrt {z^{2}+C_{1}}\right ) +C_{2}\] Using (1) in the above gives\begin {align*} bx & =\sqrt {\frac {b}{a}}\ln \left ( z+\sqrt {z^{2}+\left ( \frac {b}{a}y^{2}-z^{2}\right ) }\right ) +C_{2}\\ & =\sqrt {\frac {b}{a}}\ln \left ( z+\sqrt {\frac {b}{a}y^{2}}\right ) +C_{2}\\ C_{2} & =bx-\sqrt {\frac {b}{a}}\ln \left ( z+\sqrt {\frac {b}{a}y^{2}}\right ) \end {align*}
Since \(dw=0\) then \(w=C_{3}\). Where \(C_{1},C_{2},C_{3}\) are constants. But \(C_{3}=F\left ( C_{1},C_{2}\right ) \) where \(F\) is arbitrary function. Hence\begin {equation} u\left ( x,y,z\right ) =F\left ( \frac {b}{a}y^{2}-z^{2},bx-\sqrt {\frac {b}{a}}\ln \left ( \sqrt {\frac {b}{a}}y+z\right ) \right ) \tag {2} \end {equation} case \(dx=\frac {dy}{az}\)
From (1) we solve for \(z\) in terms of \(y\). This gives \(z^{2}=\frac {b}{a}y^{2}-C_{1}\) or \(z=\sqrt {\frac {b}{a}y^{2}-C_{1}}\), taking the positive root only. Hence\(\ dx=\frac {dy}{az}\) becomes\(\ adx=\frac {dy}{\sqrt {\frac {b}{a}y^{2}-C_{1}}}\) or \(adx=\sqrt {\frac {a}{b}}\frac {dy}{\sqrt {y^{2}-\frac {a}{b}C_{1}}}\) Integrating give \[ ax=\sqrt {\frac {a}{b}}\ln \left ( y+\sqrt {y^{2}-\frac {a}{b}C_{1}}\right ) +C_{2}\] Using (1) in the above gives\begin {align*} ax & =\sqrt {\frac {a}{b}}\ln \left ( y+\sqrt {y^{2}-\frac {a}{b}\left ( \frac {b}{a}y^{2}-z^{2}\right ) }\right ) +C_{2}\\ ax & =\sqrt {\frac {a}{b}}\ln \left ( y+\frac {a}{b}z\right ) +C_{2} \end {align*}
Hence\[ C_{2}=ax-\sqrt {\frac {a}{b}}\ln \left ( y+\sqrt {\frac {a}{b}}z\right ) \] Since \(dw=0\) then \(w=C_{3}\). Where \(C_{1},C_{2},C_{3}\) are constants. But \(C_{3}=F\left ( C_{1},C_{2}\right ) \) where \(F\) is arbitrary function. Hence\begin {equation} u\left ( x,y,z\right ) =F\left ( \frac {b}{a}y^{2}-z^{2},ax-\sqrt {\frac {a}{b}}\ln \left ( y+\frac {a}{b}z\right ) \right ) \tag {3} \end {equation} Both (2,3) are valid solutions.
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 6.2.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + a y w_y + b z w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y,z], x] + a*y*D[w[x, y,z], y] +b*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y x^{-a},z x^{-b}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y,z),x)+ a*y*diff(w(x,y,z),y) + b*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( y{x}^{-a},z{x}^{-b} \right ) \]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 6.2.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + a z w_y + b y w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y,z], x] + a*z*D[w[x, y,z], y] +b*y*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (i y \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {i \sqrt {a} z \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}},y \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {\sqrt {a} z \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y,z),x)+ a*z*diff(w(x,y,z),y) + b*y*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {{z}^{2}a-b{y}^{2}}{a}},x \left ( \sqrt {ba}y+za \right ) ^{-{\frac {1}{ba}\sqrt {ba}}} \right ) \]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 6.2.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + (a x+b y) w_y + (\alpha x+\beta y+\gamma z) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y,z], x] + (a*x+b*y)*D[w[x, y,z], y] +(alpha*x+beta*y+gamma*z)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {x^{-b} (a x+(b-1) y)}{b-1},\frac {x^{-\gamma } (-a \beta x+\alpha x (b-\gamma )-(\gamma -1) (-b z+\beta y+\gamma z))}{(\gamma -1) (b-\gamma )}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y,z),x)+ (a*x+b*y)*diff(w(x,y,z),y) + (alpha*x+beta*y+gamma*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac { \left ( y \left ( b-1 \right ) +ax \right ) {x}^{-b}}{b-1}},{\frac {- \left ( -\gamma +b \right ) \left ( a\beta -\alpha \,b+\alpha \right ) {x}^{1-\gamma }-{x}^{-\gamma } \left ( z \left ( b-1 \right ) \gamma -{b}^{2}z+ \left ( \beta \,y+z \right ) b+\beta \, \left ( ax-y \right ) \right ) \left ( -1+\gamma \right ) }{ \left ( -1+\gamma \right ) \left ( b-1 \right ) \left ( -\gamma +b \right ) }} \right ) \]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 6.2.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a b x w_x + (a y+b z) ( b w_y -a w_z)= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*b*x*D[w[x, y,z], x] + (a*y+b*z)*(b*D[w[x, y,z], y] -a*D[w[x,y,z],z])==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; pde := a*b*x*diff(w(x,y,z),x)+ (a*y+b*z)*(b*diff(w(x,y,z),y) - a*diff(w(x,y,z),z))= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay+bz}{b}},x{{\rm e}^{-{\frac {ay}{ay+bz}}}} \right ) \]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 6.2.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a b x w_x + b (a y+b z) w_y + a(a y-b z) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*b*x*D[w[x, y,z], x] + b*(a*y+b*z)*D[w[x, y,z], y] +a*(a*y-b*z)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; pde := a*b*x*diff(w(x,y,z),x)+ b*(a*y+b*z)*diff(w(x,y,z),y) + a*(a*y-b*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -{\frac {1}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}},x \left ( { \left ( {\frac {\sqrt {2}{a}^{2}y}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}+ \left ( {ay{\frac {1}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}+{zb{\frac {1}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}} \right ) \sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}} \right ) {\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}}} \right ) ^{-{\frac {a\sqrt {2}}{2}{\frac {1}{\sqrt {-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}{\frac {1}{\sqrt {{\frac {{a}^{2}}{-{a}^{2}{y}^{2}+2\,abyz+{b}^{2}{z}^{2}}}}}}}} \right ) \]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 6.2.1.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b^2 c y w_x + a^2 c x w_y - a b(a x+b y) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = b^2*c*y*D[w[x, y,z], x] + a^2*c*x*D[w[x, y,z], y] -a*b*(a*x+b*y)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {1}{2} \left (y^2-\frac {a^2 x^2}{b^2}\right ),\frac {a x+b y+c z}{c}\right )\right \}\right \}\]
Maple ✓
restart; pde := b^2*c*y*diff(w(x,y,z),x)+ a^2*c*x*diff(w(x,y,z),y) - a*b*(a*x+b*y)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-{a}^{2}{x}^{2}+{b}^{2}{y}^{2}}{{b}^{2}}},{\frac {ax+by+cz}{c}} \right ) \]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 6.2.1.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ c z w_x + (a x +b y) w_y +(a x+b y+c z) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = c*z*D[w[x, y,z], x] + (a*x+b*y)*D[w[x, y,z], y] +(a*x+b*y+c*z)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart; pde := c*z*diff(w(x,y,z),x)+ (a*x+b*y)*diff(w(x,y,z),y) + (a*x+b*y+c*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 6.2.1.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b^2 c z w_x - a^2 c x w_y + a b^2 y w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = b^2*c*z*D[w[x, y,z], x] - a^2*c*x*D[w[x, y,z], y] +a*b^2*y*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart; pde := b^2*c*z*diff(w(x,y,z),x)-a^2*c*x*diff(w(x,y,z),y) + a*b^2*y*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 6.2.1.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ (x+a) w_x + (y+b) x w_y + (z+c) w_z= 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = (x+a)*D[w[x, y,z], x] + (y+b)*D[w[x, y,z], y] +(z+c)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b+y}{a+x},\frac {c+z}{a+x}\right )\right \}\right \}\]
Maple ✓
restart; pde := (x+a)*diff(w(x,y,z),x)+(y+b)*diff(w(x,y,z),y) + (z+c)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y+b}{x+a}},{\frac {z+c}{x+a}} \right ) \]
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Added April 13, 2019.
Problem Chapter 6.2.1.14, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ 2 b c(a x-b y) w_x -a c(a x-b y-c z)w_y - a b (a x -b y-3 c z) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = 2*b*c*(a*x-b*y)*D[w[x, y,z], x] -a*c*(a*x-b*y-c*z)*D[w[x, y,z], y] - a*b*(a*x -b*y-3*c*z)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
$Aborted
Maple ✗
restart; pde := 2*b*c*(a*x-b*y)*diff(w(x,y,z),x)-a*c*(a*x-b*y-c*z)*diff(w(x,y,z),y)- a*b*(a*x -b*y-3*c*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added April 13, 2019.
Problem Chapter 6.2.1.15, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b c(y-z) w_x +a c(z-x)w_y + a b (x -y) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = b*c*(y-z)*D[w[x, y,z], x] +a*c*(z-x)*D[w[x, y,z], y] + a*b*(x -y)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
$Aborted
Maple ✓
restart; pde := b*c*(y-z)*diff(w(x,y,z),x)+a*c*(z-x)*diff(w(x,y,z),y)+ a*b*(x -y)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
\[w \left ( x,y,z \right ) ={{\rm e}^{{\frac {{x}^{2}{\it \_C2}}{2}}}}{{\rm e}^{{\it \_C1}\,x}}{{\rm e}^{{\frac {b{y}^{2}{\it \_C2}}{2\,a}}}}{{\rm e}^{{\frac {{\it \_C1}\,yb}{a}}}}{\it \_C3}\,{\it \_C5}\,{\it \_C4}\,{{\rm e}^{{\frac {c{z}^{2}{\it \_C2}}{2\,a}}}}{{\rm e}^{{\frac {cz{\it \_C1}}{a}}}}\]
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Added April 13, 2019.
Problem Chapter 6.2.1.16, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ b c(b y-2 c z) w_x +a c(3 c z-a x) w_y + a b (2 a x -3 b y) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = b*c*(b*y-2*c*z)*D[w[x, y,z], x] +a*c*(3*c*z-a*x)*D[w[x, y,z], y] + a*b*(2*a*x -3*b*y)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
$Aborted
Maple ✓
restart; pde := b*c*(b*y-2*c*z)*diff(w(x,y,z),x)+a*c*(3*c*z-a*x)*diff(w(x,y,z),y)+ a*b*(2*a*x-3*b*y)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
\[w \left ( x,y,z \right ) ={{\rm e}^{{\frac {{x}^{2}{\it \_C2}}{2}}}}{{\rm e}^{{\it \_C1}\,x}}{{\rm e}^{{\frac {{b}^{2}{\it \_C2}\,{y}^{2}}{2\,{a}^{2}}}}}{{\rm e}^{{\frac {2\,{\it \_C1}\,yb}{3\,a}}}}{\it \_C3}\,{\it \_C5}\,{\it \_C4}\,{{\rm e}^{{\frac {{c}^{2}{\it \_C2}\,{z}^{2}}{2\,{a}^{2}}}}}{{\rm e}^{{\frac {cz{\it \_C1}}{3\,a}}}}\]
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Added April 13, 2019.
Problem Chapter 6.2.1.17, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ 2 b c(b y-c z) w_x -a c(4 a x-3 b y-c z) w_y + 3 a b (4 a x-b y-3 c z) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = 2*b*c*(b*y-c*z)*D[w[x, y,z], x] -a*c*(4*a*x-3*b*y-c*z)*D[w[x, y,z], y] + 3*a*b*(4*a*x-b*y-3*c*z)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
$Aborted
Maple ✗
restart; pde := 2*b*c*(b*y-c*z)*diff(w(x,y,z),x)-a*c*(4*a*x-3*b*y-c*z)*diff(w(x,y,z),y)+ 3*a*b*(4*a*x-b*y-3*c*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added April 13, 2019.
Problem Chapter 6.2.1.18, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ (a x+y-z) w_x -(x+a y-z) w_y + (a-1) (y-x) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x+y-z)*D[w[x, y,z], x] -(x+a*y-z)*D[w[x, y,z], y] + (a-1)*(y-x)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
$Aborted
Maple ✗
restart; pde := (a*x+y-z)*diff(w(x,y,z),x)-(x+a*y-z)*diff(w(x,y,z),y)+ (a-1)*(y-x)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added April 13, 2019.
Problem Chapter 6.2.1.19, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ 2 b c (3 a x-2 b y+c z) w_x -2 a c(2 a x-5 b y+3 c z) w_y + a b(2 a x-6 b y+11 c z) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = 2*b*c*(3*a*x-2*b*y+c*z)*D[w[x, y,z], x] -2*a*c(2*a*x-5*b*y+3*c*z)*D[w[x, y,z], y] + a*b(2*a*x-6*b*y+11*c*z)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart; pde := 2*b*c*(3*a*x-2*b*y+c*z)*diff(w(x,y,z),x)-2*a*c*(2*a*x-5*b*y+3*c*z)*diff(w(x,y,z),y)+ a*b*(2*a*x-6*b*y+11*c*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added April 13, 2019.
Problem Chapter 6.2.1.20, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ (A x+c y+b z) w_x +(c x+ B y+a z) w_y + (b x + a y + C z) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (A*x+c*y+b*z)*D[w[x, y,z], x] +(c*x+B*y+a*z)*D[w[x, y,z], y] +(b*x+a*y+C1*z)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart; pde := (A*x+c*y+b*z)*diff(w(x,y,z),x)+(c*x+B*y+a*z)*diff(w(x,y,z),y)+ (b*x+a*y+C1*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added April 13, 2019.
Problem Chapter 6.2.1.21, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ (a_1 x+b_1 y+c_1 z+d_1) w_x +(a_2 x+b_2 y+c_2 z+d_2) w_y + (a_3 x+b_3 y+c_3 z+d_3) w_z= 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a1*x+b1*y+c1*z+d1)*D[w[x, y,z], x] +(a2*x+b2*y+c2*z+d2)*D[w[x, y,z], y] +(a3*x+b3*y+c3*z+d3)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart; pde := (a1*x+b1*y+c1*z+d1)*diff(w(x,y,z),x)+(a2*x+b2*y+c2*z+d2)*diff(w(x,y,z),y)+ (a3*x+b3*y+c3*z+d3)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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