6.7.1 2.1

6.7.1.1 [1579] Problem 1

problem number 1579

Added May 31, 2019.

Problem Chapter 7.2.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b w_y + c w_z = \alpha x+\beta y+\gamma z + \delta \]

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]==alpha*x+beta*y+gamma*z+delta; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*diff(w(x,y,z),z)= alpha*x+beta*y+gamma*z+delta; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.7.1.2 [1580] Problem 2

problem number 1580

Added May 31, 2019.

Problem Chapter 7.2.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + a z w_y + b y w_z = c x+s \]

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]==c*x+s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {b}},\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {a}}\right )+\frac {c x^2}{2}+s x\right \}\right \}\]

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)= c*x+s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {f_{1} \left (\frac {z^{2} a -b \,y^{2}}{a}, \frac {x \sqrt {a b}-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right ) \sqrt {a b}-\int _{}^{y}-\frac {\left (c x +s \right ) \sqrt {a b}-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right ) c +\ln \left (\frac {a b \textit {\_a} +\sqrt {a b}\, \sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +z^{2} a \right )}}{\sqrt {a b}}\right ) c}{\sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +z^{2} a \right )}}d \textit {\_a}}{\sqrt {a b}}\]

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6.7.1.3 [1581] Problem 3

problem number 1581

Added May 31, 2019.

Problem Chapter 7.2.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + (a_1 x + a_0) w_y + (b_1 x+b_0) w_z = \alpha x+\beta y+\gamma z + \delta \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a1*x+a0)*D[w[x, y,z], y] +(b1*x+b0)*D[w[x,y,z],z]==alpha*x+beta*y+gamma*z+delta; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \frac {1}{6} x \left (-3 \text {a0} \beta x-2 \text {a1} \beta x^2+3 \alpha x-3 \text {b0} \gamma x-2 \text {b1} \gamma x^2+6 \beta y+6 \delta +6 \gamma z\right )+c_1\left (-\text {a0} x-\frac {\text {a1} x^2}{2}+y,-\text {b0} x-\frac {\text {b1} x^2}{2}+z\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+  (a1*x+a0)*diff(w(x,y,z),y)+(b1*x+b0)*diff(w(x,y,z),z)=alpha*x+beta*y+gamma*z+delta; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-\frac {1}{2} \operatorname {a1} \,x^{2}-\operatorname {a0} x +y , -\frac {1}{2} \operatorname {b1} \,x^{2}-\operatorname {b0} x +z \right )-\frac {\left (\beta \operatorname {a1} +\operatorname {b1} \gamma \right ) x^{3}}{3}+\frac {\left (-\operatorname {a0} \beta -\gamma \operatorname {b0} +\alpha \right ) x^{2}}{2}+\left (\beta y +\gamma z +\delta \right ) x\]

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6.7.1.4 [1582] Problem 4

problem number 1582

Added May 31, 2019.

Problem Chapter 7.2.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + (a_2 y + a_1 x + a_0) w_y + (b_2 y+b_1 x + b_0) w_z = c_2 y+c_1 z+c_0 x+s \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a2*y+a1*x+a0)*D[w[x, y,z], y] +(b2*y+b1*x+b0)*D[w[x,y,z],z]==c2*y+c1*z+c0*x+s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {\text {a2} s+\text {b2} \text {c1} e^{\text {a2} (K[3]-x)} y+\text {a2} \text {c2} e^{\text {a2} (K[3]-x)} y-\text {b2} \text {c1} y+\text {a2} \text {c1} z+\text {a2} \text {c0} K[3]-\left (\text {a2} e^{\text {a2} K[3]} \text {c2}+\text {b2} \text {c1} \left (-1+e^{\text {a2} K[3]}\right )\right ) \int _1^xe^{-\text {a2} K[1]} (\text {a0}+\text {a1} K[1])dK[1]+\left (\text {a2} e^{\text {a2} K[3]} \text {c2}+\text {b2} \text {c1} \left (-1+e^{\text {a2} K[3]}\right )\right ) \int _1^{K[3]}e^{-\text {a2} K[1]} (\text {a0}+\text {a1} K[1])dK[1]-\text {a2} \text {c1} \int _1^x\left (\text {b0}+\text {b1} K[2]+\frac {\text {b2} \left (-1+e^{-\text {a2} K[2]}\right ) (\text {a0}+\text {a1} K[2])}{\text {a2}}\right )dK[2]+\text {a2} \text {c1} \int _1^{K[3]}\left (\text {b0}+\text {b1} K[2]+\frac {\text {b2} \left (-1+e^{-\text {a2} K[2]}\right ) (\text {a0}+\text {a1} K[2])}{\text {a2}}\right )dK[2]}{\text {a2}}dK[3]+c_1\left (y e^{-\text {a2} x}-\int _1^xe^{-\text {a2} K[1]} (\text {a0}+\text {a1} K[1])dK[1],-\int _1^x\left (\text {b0}+\text {b1} K[2]+\frac {\text {b2} \left (-1+e^{-\text {a2} K[2]}\right ) (\text {a0}+\text {a1} K[2])}{\text {a2}}\right )dK[2]+\frac {\text {b2} y \left (e^{-\text {a2} x}-1\right )}{\text {a2}}+z\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+  (a2*y+a1*x+a0)*diff(w(x,y,z),y)+(b2*y+b1*x+b0)*diff(w(x,y,z),z)=c2*y+c1*z+c0*x+s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {6 f_{1} \left (\frac {\left (y \,\operatorname {a2}^{2}+\operatorname {a2} \left (\operatorname {a1} x +\operatorname {a0} \right )+\operatorname {a1} \right ) {\mathrm e}^{-\operatorname {a2} x}}{\operatorname {a2}^{2}}, \frac {\left (-\operatorname {b1} \,x^{2}-2 \operatorname {b0} x +2 z \right ) \operatorname {a2}^{3}+\operatorname {b2} \left (\operatorname {a1} \,x^{2}+2 \operatorname {a0} x -2 y \right ) \operatorname {a2}^{2}-2 \operatorname {a0} \operatorname {a2} \operatorname {b2} -2 \operatorname {a1} \operatorname {b2}}{2 \operatorname {a2}^{3}}\right ) \operatorname {a2}^{4}+6 \left (-\frac {\operatorname {c1} \operatorname {b1} \,x^{2}}{3}+\frac {\left (-\operatorname {c1} \operatorname {b0} +\operatorname {c0} \right ) x}{2}+\operatorname {c1} z +s \right ) x \,\operatorname {a2}^{4}+\left (2 \operatorname {b2} \operatorname {c1} \operatorname {a1} \,x^{3}+3 \left (\operatorname {b2} \operatorname {c1} \operatorname {a0} -\operatorname {a1} \operatorname {c2} \right ) x^{2}-6 \left (\operatorname {b2} \operatorname {c1} y +\operatorname {c2} \operatorname {a0} \right ) x +6 \operatorname {c2} y \right ) \operatorname {a2}^{3}+3 \left (-\operatorname {c1} \operatorname {a1} \operatorname {b2} \,x^{2}-2 \operatorname {c1} \operatorname {a0} \operatorname {b2} x +2 \operatorname {b2} \operatorname {c1} y +2 \operatorname {c2} \operatorname {a0} \right ) \operatorname {a2}^{2}+6 \left (\operatorname {b2} \operatorname {c1} \operatorname {a0} +\operatorname {a1} \operatorname {c2} \right ) \operatorname {a2} +6 \operatorname {b2} \operatorname {c1} \operatorname {a1}}{6 \operatorname {a2}^{4}}\]

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6.7.1.5 [1583] Problem 5

problem number 1583

Added May 31, 2019.

Problem Chapter 7.2.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ w_x + (a y + k_1 x + k_0) w_y + (b z+s_1 x + s_0) w_z = c_1 x+c_0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a*y+k1*x+k0)*D[w[x, y,z], y] +(b*z+s1*x+s0)*D[w[x,y,z],z]==c1*x+c0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} (\text {k0}+\text {k1} K[1])dK[1],z e^{-b x}-\int _1^xe^{-b K[2]} (\text {s0}+\text {s1} K[2])dK[2]\right )+\text {c0} x+\frac {\text {c1} x^2}{2}\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+  (a*y+k1*x+k0)*diff(w(x,y,z),y)+(b*z+s1*x+s0)*diff(w(x,y,z),z)=c1*x+c0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {\operatorname {c1} \,x^{2}}{2}+\operatorname {c0} x +f_{1} \left (\frac {{\mathrm e}^{-a x} \left (y \,a^{2}+a \left (\operatorname {k1} x +\operatorname {k0} \right )+\operatorname {k1} \right )}{a^{2}}, \frac {{\mathrm e}^{-b x} \left (z \,b^{2}+b \left (\operatorname {s1} x +\operatorname {s0} \right )+\operatorname {s1} \right )}{b^{2}}\right )\]

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6.7.1.6 [1584] Problem 6

problem number 1584

Added May 31, 2019.

Problem Chapter 7.2.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b y w_y + c z w_z = \alpha x+\beta y+\gamma z + \delta \]

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]==alpha*x+beta*y+gamma*z+delta; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
pde := a*diff(w(x,y,z),x)+  b*y*diff(w(x,y,z),y)+c*z*diff(w(x,y,z),z)=alpha*x+beta*y+gamma*z+delta; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

Hand solution

Solve

\[ aw_{x}+byw_{y}+czw_{z}=\alpha x+\beta y+\gamma z+\delta \]

The parametrization invariant Lagrange-Charpit equations are

\[ \frac {dx}{a}=\frac {dy}{by}=\frac {dz}{cz}=\frac {dw}{\alpha x+\beta y+\gamma z+\delta }\]

Solving \(\frac {dx}{a}=\frac {dy}{by}\) gives

\begin{align} \frac {b}{a}dx & =\frac {dy}{y}\nonumber \\ \frac {b}{a}x & =\ln y+C_{1}\tag {1}\\ \ln y & =\frac {b}{a}x-C_{1}\nonumber \\ y & =C_{1}e^{\frac {b}{a}x}\nonumber \\ C_{1} & =ye^{-\frac {b}{a}x}\nonumber \end{align}

Equation \(\frac {dx}{a}=\frac {dz}{cz}\) gives

\begin{align} \frac {c}{a}dx & =\frac {dz}{z}\nonumber \\ \frac {c}{a}x & =\ln z+C_{2}\nonumber \\ \ln z & =\frac {c}{a}x-C_{2}\nonumber \\ z & =C_{2}e^{\frac {c}{a}x}\nonumber \\ C_{2} & =ze^{-\frac {c}{a}x}\tag {2}\end{align}

And \(\frac {dx}{a}=\frac {dw}{\alpha x+\beta y+\gamma z+\delta }\) gives

\begin{align*} \frac {\alpha x+\beta y+\gamma z+\delta }{a}dx & =dw\\ \left ( \frac {\alpha }{a}x+\beta \frac {y}{a}+\gamma \frac {z}{a}+\frac {\delta }{a}\right ) dx & =dw \end{align*}

But from (1) \(y=C_{1}e^{\frac {b}{a}x}\) and from (2) \(z=C_{2}e^{\frac {c}{a}x}\). Hence the above becomes

\[ \left ( \frac {\alpha }{a}x+\frac {\beta }{a}C_{1}e^{\frac {b}{a}x}+\frac {\gamma }{a}C_{2}e^{\frac {c}{a}x}+\frac {\delta }{a}\right ) dx=dw \]

Integrating

\[ \frac {\alpha }{a}\frac {x^{2}}{2}+\frac {\beta }{b}C_{1}e^{\frac {b}{a}x}+\frac {\gamma }{c}C_{2}e^{\frac {c}{a}x}+\frac {\delta }{a}x=w+C_{3}\]

But \(C_{2}=ze^{-\frac {c}{a}x}\) and \(C_{1}=ye^{-\frac {b}{a}x}\), hence the above becomes

\begin{align*} \frac {\alpha }{a}\frac {x^{2}}{2}+\frac {\beta }{b}ye^{-\frac {b}{a}x}e^{\frac {b}{a}x}+\frac {\gamma }{c}ze^{-\frac {c}{a}x}e^{\frac {c}{a}x}+\frac {\delta }{a}x & =w+C_{3}\\ \frac {\alpha }{a}\frac {x^{2}}{2}+\frac {\beta }{b}y+\frac {\gamma }{c}z+\frac {\delta }{a}x & =w+C_{3}\\ C_{3} & =\left ( \frac {\alpha }{a}\frac {x^{2}}{2}+\frac {\beta }{b}y+\frac {\gamma }{c}z+\frac {\delta }{a}x\right ) -w \end{align*}

Since \(C_{3}=F\left ( C_{1},C_{2}\right ) \) then the solution is

\begin{align*} \left ( \frac {\alpha }{a}\frac {x^{2}}{2}+\frac {\beta }{b}y+\frac {\gamma }{c}z+\frac {\delta }{a}x\right ) -w & =F\left ( ye^{-\frac {b}{a}x},ze^{-\frac {c}{a}x}\right ) \\ w\left ( x,y,z\right ) & =F\left ( ye^{-\frac {b}{a}x},ze^{-\frac {c}{a}x}\right ) +\left ( \frac {\alpha }{a}\frac {x^{2}}{2}+\frac {\beta }{b}y+\frac {\gamma }{c}z+\frac {\delta }{a}x\right ) \end{align*}

(sign change on \(F\) does not matter, since arbitrary function, can be renamed).

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6.7.1.7 [1585] Problem 7

problem number 1585

Added June 1, 2019.

Problem Chapter 7.2.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ x w_x + a z w_y + b y w_z = c \]

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]==c; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c \log (x)+c_1\left (i y \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {i \sqrt {a} z \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}},y \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {\sqrt {a} z \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}}\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := x*diff(w(x,y,z),x)+  a*z*diff(w(x,y,z),y)+b*y*diff(w(x,y,z),z)=c; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

Hand solution

Solve

\[ xw_{x}+azw_{y}+byw_{z}=c \]

The parametrization invariant Lagrange-Charpit equations are

\[ \frac {dx}{x}=\frac {dy}{az}=\frac {dz}{by}=\frac {dw}{c}\]

Solving \(\frac {dy}{az}=\frac {dz}{by}\) gives

\begin{align} \frac {b}{a}ydy & =zdz\nonumber \\ \frac {b}{a}y^{2} & =z^{2}+C_{1}\tag {1}\\ C_{1} & =\frac {b}{a}y^{2}-z^{2}\nonumber \\ & =\frac {by^{2}-az^{2}}{a}\nonumber \end{align}

Equation \(\frac {dx}{x}=\frac {dy}{az}\) gives

\[ a\frac {dx}{x}=\frac {dy}{z}\]

But from (1) \(z=\sqrt {\frac {b}{a}y^{2}-C_{1}}\), hence the above becomes

\[ a\frac {dx}{x}=\frac {dy}{\sqrt {\frac {b}{a}y^{2}-C_{1}}}\]

Integrating gives

\begin{align*} a\ln x & =\sqrt {\frac {a}{b}}\ln \left ( \sqrt {\frac {b}{a}}y+\sqrt {\frac {b}{a}y^{2}-C_{1}}\right ) +C_{2}\\ \ln x & =\sqrt {\frac {1}{ab}}\ln \left ( \sqrt {\frac {b}{a}}y+\sqrt {\frac {b}{a}y^{2}-C_{1}}\right ) +\frac {C_{2}}{a}\end{align*}

Let \(\frac {C_{2}}{a}=C_{3}\) and the above becomes

\begin{align*} x & =C_{3}\left ( \sqrt {\frac {b}{a}}y+\sqrt {\frac {b}{a}y^{2}-C_{1}}\right ) ^{\sqrt {\frac {1}{ab}}}\\ C_{3} & =x\left ( \sqrt {\frac {b}{a}}y+\sqrt {\frac {b}{a}y^{2}-C_{1}}\right ) ^{-\sqrt {\frac {1}{ab}}}\end{align*}

But from (1)

\begin{equation} C_{1}=\frac {b}{a}y^{2}-z^{2}\tag {2}\end{equation}

Hence \(C_{3}\) simplifies to

\begin{align} C_{3} & =x\left ( \sqrt {\frac {b}{a}}y+\sqrt {\frac {b}{a}y^{2}-\left ( \frac {b}{a}y^{2}-z^{2}\right ) }\right ) ^{-\sqrt {\frac {1}{ab}}}\nonumber \\ & =x\left ( \sqrt {\frac {b}{a}}y+z\right ) ^{-\sqrt {\frac {1}{ab}}}\tag {4}\end{align}

And \(\frac {dx}{x}=\frac {dw}{c}\) gives

\[ \ln x=\frac {1}{c}w+C_{4}\]

But \(C_{4}=F\left ( C_{1},C_{3}\right ) \). Hence

\begin{align*} \ln x-\frac {1}{c}w & =F\left ( \frac {b}{a}y^{2}-z^{2},x\left ( \sqrt {\frac {b}{a}}y+z\right ) ^{-\sqrt {\frac {1}{ab}}}\right ) \\ -\frac {1}{c}w & =F\left ( \frac {b}{a}y^{2}-z^{2},x\left ( \sqrt {\frac {b}{a}}y+z\right ) ^{-\sqrt {\frac {1}{ab}}}\right ) -\ln x\\ w\left ( x,y,z\right ) & =cF\left ( \frac {b}{a}y^{2}-z^{2},x\left ( \sqrt {\frac {b}{a}}y+z\right ) ^{-\sqrt {\frac {1}{ab}}}\right ) +c\ln x \end{align*}

Verified OK under the assumptions that \(a>0,b>0,z>0\).

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6.7.1.8 [1586] Problem 8

problem number 1586

Added June 1, 2019.

Problem Chapter 7.2.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a b x w_x + b(a y+b z) w_y + a(a y-b z) w_z = c \]

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]==c; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
local gamma; 
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)=c; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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6.7.1.9 [1587] Problem 9

problem number 1587

Added June 1, 2019.

Problem Chapter 7.2.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ (a_1 x+a_0) w_x + (b_1 y+b_0) w_y + (c_1 z+c_0) w_z = \alpha x+\beta y+\gamma z + \delta \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (a1*x+a0)*D[w[x, y,z], x] + (b1*y+b0)*D[w[x, y,z], y] +(c1*z+c0)*D[w[x,y,z],z]==alpha*x+beta*y+gamma*z+delta; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {(\text {b0}+\text {b1} y) (\text {a0}+\text {a1} x)^{-\frac {\text {b1}}{\text {a1}}}}{\text {b1}},\frac {(\text {c0}+\text {c1} z) (\text {a0}+\text {a1} x)^{-\frac {\text {c1}}{\text {a1}}}}{\text {c1}}\right )-\frac {\text {a0} \alpha \log (\text {a0}+\text {a1} x)}{\text {a1}^2}+\frac {\log (\text {a0}+\text {a1} x) (-\text {b0} \beta \text {c1}-\text {b1} \text {c0} \gamma +\text {b1} \text {c1} \delta )}{\text {a1} \text {b1} \text {c1}}+\frac {\alpha x}{\text {a1}}+\frac {\text {b0} \beta }{\text {b1}^2}+\frac {\beta y}{\text {b1}}+\frac {\gamma (\text {c0}+\text {c1} z)}{\text {c1}^2}\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := (a1*x+a0)*diff(w(x,y,z),x)+  (b1*y+b0)*diff(w(x,y,z),y)+(c1*z+c0)*diff(w(x,y,z),z)=alpha*x+beta*y+gamma*z+delta; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {\alpha x}{\operatorname {a1}}+\frac {\beta y}{\operatorname {b1}}+\frac {\gamma z}{\operatorname {c1}}+\frac {\operatorname {b0} \beta }{\operatorname {b1}^{2}}+\frac {\gamma \operatorname {c0}}{\operatorname {c1}^{2}}-\frac {\left (\left (\left (\gamma \operatorname {c0} -\operatorname {c1} \delta \right ) \operatorname {b1} +\operatorname {b0} \beta \operatorname {c1} \right ) \operatorname {a1} +\operatorname {a0} \alpha \operatorname {b1} \operatorname {c1} \right ) \ln \left (\operatorname {a1} x +\operatorname {a0} \right )}{\operatorname {a1}^{2} \operatorname {b1} \operatorname {c1}}+f_{1} \left (\frac {\left (\operatorname {b1} y +\operatorname {b0} \right ) \left (\operatorname {a1} x +\operatorname {a0} \right )^{-\frac {\operatorname {b1}}{\operatorname {a1}}}}{\operatorname {b1}}, \frac {\left (\operatorname {c1} z +\operatorname {c0} \right ) \left (\operatorname {a1} x +\operatorname {a0} \right )^{-\frac {\operatorname {c1}}{\operatorname {a1}}}}{\operatorname {c1}}\right )\]

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