6.6.1 2.1

6.6.1.1 [1367] Problem 1

problem number 1367

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y a -x b}{a}, \frac {z a -x c}{a}\right )\]

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

\[ u\left ( x,y,z\right ) =F\left ( \frac {b}{a}x-y,\frac {c}{a}x-z\right ) \]

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6.6.1.2 [1368] Problem 2

problem number 1368

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-\frac {a \,x^{2}}{2}+y , \frac {x \left (a \,x^{2}-3 y \right ) b}{3}+z \right )\]

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*}

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6.6.1.3 [1369] Problem 3

problem number 1369

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (y \,{\mathrm e}^{-\frac {b x}{a}}, z \,{\mathrm e}^{-\frac {c x}{a}}\right )\]

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

\[ u\left ( x,y,z\right ) =F\left ( ye^{-abx},ze^{-\frac {c}{a}x}\right ) \]

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6.6.1.4 [1370] Problem 4

problem number 1370

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {z^{2} a -b \,y^{2}}{a}, \frac {x \sqrt {a b}-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )\]

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.

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6.6.1.5 [1371] Problem 5

problem number 1371

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (y \,x^{-a}, z \,x^{-b}\right )\]

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6.6.1.6 [1372] Problem 6

problem number 1372

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {z^{2} a -b \,y^{2}}{a}, x \left (\sqrt {a b}\, y +a z \right )^{-\frac {\sqrt {a b}}{a b}}\right )\]

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6.6.1.7 [1373] Problem 7

problem number 1373

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {\left (\left (b -1\right ) y +a x \right ) x^{-b}}{b -1}, -\frac {\left (z \,\gamma ^{2}+\left (\alpha x -z b +\beta y -z \right ) \gamma +\left (-\alpha x +z \right ) b +\beta \left (a x -y \right )\right ) x^{-\gamma }}{\left (\gamma -1\right ) \left (b -\gamma \right )}\right )\]

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6.6.1.8 [1374] Problem 8

problem number 1374

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y +b z}{b}, x \,{\mathrm e}^{-\frac {a y}{a y +b z}}\right )\]

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6.6.1.9 [1375] Problem 9

problem number 1375

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {x^{\sqrt {2}} \left (\left (\sqrt {2}-2\right ) a y+\sqrt {2} b z\right )}{4 a},\frac {x^{-\sqrt {2}} \left (\left (2+\sqrt {2}\right ) a y+\sqrt {2} b z\right )}{4 a}\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-\frac {1}{\sqrt {-a^{2} y^{2}+2 a b y z +b^{2} z^{2}}}, x {\left (-\frac {\left (a^{3} y^{3}-a^{2} b \,y^{2} z -3 a \,b^{2} y \,z^{2}-b^{3} z^{3}\right ) \sqrt {-\frac {a^{2}}{a^{2} y^{2}-2 a b y z -b^{2} z^{2}}}-a^{2} y \sqrt {-2 a^{2} y^{2}+4 a b y z +2 b^{2} z^{2}}}{\sqrt {-\frac {a^{2}}{a^{2} y^{2}-2 a b y z -b^{2} z^{2}}}\, \left (-a^{2} y^{2}+2 a b y z +b^{2} z^{2}\right )^{{3}/{2}}}\right )}^{-\frac {a}{\sqrt {-2 a^{2} y^{2}+4 a b y z +2 b^{2} z^{2}}\, \sqrt {-\frac {a^{2}}{a^{2} y^{2}-2 a b y z -b^{2} z^{2}}}}}\right )\]

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6.6.1.10 [1376] Problem 10

problem number 1376

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-a^{2} x^{2}+y^{2} b^{2}}{b^{2}}, \frac {a x +b y +z c}{c}\right )\]

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6.6.1.11 [1377] Problem 11

problem number 1377

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=()

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6.6.1.12 [1378] Problem 12

problem number 1378

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=()

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6.6.1.13 [1379] Problem 13

problem number 1379

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {y +b}{x +a}, \frac {z +c}{x +a}\right )\]

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6.6.1.14 [1380] Problem 14

problem number 1380

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]];
 

\begin{align*}& \left \{w(x,y,z)\to c_1\left (\frac {-a x-3 b y+c z}{c},-\frac {a x-b y+c z}{a \sqrt {-4 a x+4 b y+4 c z}}\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (\frac {-a x-3 b y+c z}{c},\frac {a x-b y+c z}{a \sqrt {-4 a x+4 b y+4 c z}}\right )\right \}\\\end{align*}

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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6.6.1.15 [1381] Problem 15

problem number 1381

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a x+b y+c z}{c},\frac {-a^2 x^2+a x (c (x-2 z)-2 b y)+b y (c (y-2 z)-b y)}{b (b+c)}\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = c_{3} c_{4} c_{5} {\mathrm e}^{\frac {\left (2 x a +2 b y +2 c z \right ) c_{1} +c_{2} \left (a \,x^{2}+b \,y^{2}+c \,z^{2}\right )}{2 a}}\]

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6.6.1.16 [1382] Problem 16

problem number 1382

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {3 a x+2 b y+c z}{c},-\frac {8 a^2 x^2+6 a x (2 b y+c z)+b y (3 b y+4 c z)}{5 b^2}\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = c_{3} c_{4} c_{5} {\mathrm e}^{\frac {3 a^{2} c_{2} x^{2}+3 b^{2} c_{2} y^{2}+3 c^{2} c_{2} z^{2}+6 a^{2} c_{1} x +4 a b c_{1} y +2 a c c_{1} z}{6 a^{2}}}\]

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6.6.1.17 [1383] Problem 17

problem number 1383

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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6.6.1.18 [1384] Problem 18

problem number 1384

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]];
 

\[\{\{w(x,y,z)\to c_1(x+y+z,(a-1) x y-x z-y z)\}\}\]

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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6.6.1.19 [1385] Problem 19

problem number 1385

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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6.6.1.20 [1386] Problem 20

problem number 1386

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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6.6.1.21 [1387] Problem 21

problem number 1387

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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