6.6.23 8.1

6.6.23.1 [1551] Problem 1

problem number 1551

Added May 31, 2019.

Problem Chapter 6.8.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + f(x) w_y + g(x) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + f[x]*D[w[x, y,z], y] +g[x]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\int _1^xf(K[1])dK[1],z-\int _1^xg(K[2])dK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+ f(x)*diff(w(x,y,z),y)+g(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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!f \left ( x \right ) \,{\rm d}x+y,-\int \!g \left ( x \right ) \,{\rm d}x+z \right ) \]

____________________________________________________________________________________

6.6.23.2 [1552] Problem 2

problem number 1552

Added May 31, 2019.

Problem Chapter 6.8.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + f(x) (y+a) w_y + g(x) (z+b) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + f[x]*(y+a)*D[w[x, y,z], y] +g[x]*(z+b)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y \exp \left (-\int _1^xf(K[1])dK[1]\right )-\int _1^xa \exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right ) f(K[2])dK[2],z \exp \left (-\int _1^xg(K[3])dK[3]\right )-\int _1^xb \exp \left (-\int _1^{K[4]}g(K[3])dK[3]\right ) g(K[4])dK[4]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+ f(x)*(y+a)*diff(w(x,y,z),y)+g(x)*(z+b)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( \left ( y+a \right ) {{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}}, \left ( z+b \right ) {{\rm e}^{-\int \!g \left ( x \right ) \,{\rm d}x}} \right ) \]

____________________________________________________________________________________

6.6.23.3 [1553] Problem 3

problem number 1553

Added May 31, 2019.

Problem Chapter 6.8.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y + f(x)) w_y + (b z+g(x)) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a*y+f[x])*D[w[x, y,z], y] +(b*z+g[x])*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} f(K[1])dK[1],z e^{-b x}-\int _1^xe^{-b K[2]} g(K[2])dK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+ (a*y+f(x))*diff(w(x,y,z),y)+(b*z+g(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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!f \left ( x \right ) {{\rm e}^{-ax}}\,{\rm d}x+y{{\rm e}^{-ax}},-\int \!g \left ( x \right ) {{\rm e}^{-bx}}\,{\rm d}x+z{{\rm e}^{-bx}} \right ) \]

____________________________________________________________________________________

6.6.23.4 [1554] Problem 4

problem number 1554

Added May 31, 2019.

Problem Chapter 6.8.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (f_1(x) y + f_2(x)) w_y + (g_1(x) y+g_2(x)) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = D[w[x, y,z], x] + (f1[x]*y+f2[x])*D[w[x, y,z], y] +(g1[x]*y+g2[x])*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3],-\int _1^x\left (\text {g2}(K[4])-\exp \left (-\int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[4]) \int _1^{K[4]}\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]\right )dK[4]-y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) \int _1^x\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]+z\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+ (f1(x)*y+f2(x))*diff(w(x,y,z),y)+(g1(x)*y+g2(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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}},-\int ^{x}\!{{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}y{\it g1} \left ( {\it \_f} \right ) -{{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it g1} \left ( {\it \_f} \right ) \int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it g1} \left ( {\it \_f} \right ) \int \!{\it f2} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it f1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}+{\it g2} \left ( {\it \_f} \right ) {d{\it \_f}}+z \right ) \]

____________________________________________________________________________________

6.6.23.5 [1555] Problem 5

problem number 1555

Added May 31, 2019.

Problem Chapter 6.8.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (f_1(x) y + f_2(x)) w_y + (g_1(x) z+g_2(x)) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (f1[x]*y+f2[x])*D[w[x, y,z], y] +(g1[x]*z+g2[x])*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2],z \exp \left (-\int _1^x\text {g1}(K[3])dK[3]\right )-\int _1^x\exp \left (-\int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+ (f1(x)*y+f2(x))*diff(w(x,y,z),y)+(g1(x)*z+g2(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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!{\it f2} \left ( x \right ) {{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it f1} \left ( x \right ) \,{\rm d}x}},-\int \!{\it g2} \left ( x \right ) {{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+z{{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) \]

____________________________________________________________________________________

6.6.23.6 [1556] Problem 6

problem number 1556

Added May 31, 2019.

Problem Chapter 6.8.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (f_2(x) y + f_1(x) z+f_0(x)) w_y + (g_2(x) y+g_1(x) z + g_0(x)) w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde = D[w[x, y,z], x] + (f2[x]*y+f1[x]*z+f0[x])*D[w[x, y,z], y] +(g2[x]*y+g1[x]*z+g0[x])*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 :=  diff(w(x,y,z),x)+ (f2(x)*y+f1(x)*z+f0(x))*diff(w(x,y,z),y)+(g2(x)*y+g1(x)*z+g0(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=()

____________________________________________________________________________________

6.6.23.7 [1557] Problem 7

problem number 1557

Added May 31, 2019.

Problem Chapter 6.8.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (y^2-a^2+a \lambda \sinh (\lambda x)-a^2 \sinh ^2(\lambda x)) w_y + f(x) \sinh (\gamma z) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (y^2-a^2+a*lambda*Sinh[lambda*x]-a^2*Sinh[lambda*x]^2)*D[w[x, y,z], y] +f[x]*Sinh[gamma*z]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log \left (\tanh \left (\frac {\gamma z}{2}\right )\right )}{\gamma }-\int _1^xf(K[2])dK[2],\frac {2 \lambda e^{\frac {a e^{-\lambda x} \left (e^{2 \lambda x}-1\right )}{\lambda }+\lambda x}}{a e^{2 \lambda x}+a-2 y e^{\lambda x}}-\int _1^{e^{\lambda x}}\frac {e^{\frac {a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]}dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+ (y^2-a^2+a*lambda*sinh(lambda*x)-a^2*sinh(lambda*x)^2)*diff(w(x,y,z),y)+f(x)*sinh(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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -2\,{\sqrt {\sinh \left ( \lambda \,x \right ) +i} \left ( \left ( i \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}-i+2\,\sinh \left ( \lambda \,x \right ) \right ) \left ( a\cosh \left ( \lambda \,x \right ) +y \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) -\lambda \,\HeunCPrime \left ( {\frac {4\,ia}{\lambda }},-1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \left ( i\sinh \left ( \lambda \,x \right ) -1/2\, \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}+1/2 \right ) \cosh \left ( \lambda \,x \right ) \right ) \left ( \left ( \left ( 2\,i \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{3}a+ \left ( i\lambda +2\,a \right ) \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}+ \left ( 2\,ia+2\,\lambda \right ) \sinh \left ( \lambda \,x \right ) -i\lambda +2\,a \right ) \cosh \left ( \lambda \,x \right ) +2\, \left ( 1+i\sinh \left ( \lambda \,x \right ) \right ) y \left ( \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}+1 \right ) \right ) \HeunC \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) -\lambda \, \left ( -\sinh \left ( \lambda \,x \right ) +i \right ) \left ( \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{2}+1 \right ) \HeunCPrime \left ( {\frac {4\,ia}{\lambda }},1/2,-1/2,{\frac {2\,ia}{\lambda }},-1/8\,{\frac {8\,ia-3\,\lambda }{\lambda }},1/2-i/2\sinh \left ( \lambda \,x \right ) \right ) \cosh \left ( \lambda \,x \right ) \right ) ^{-1}},-\int \!f \left ( x \right ) \,{\rm d}x-16\,\arctanh \left ( {{\rm e}^{z/8}} \right ) \right ) \]

____________________________________________________________________________________

6.6.23.8 [1558] Problem 8

problem number 1558

Added May 31, 2019.

Problem Chapter 6.8.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (f_1(x) y+f_2(x) y^k) w_y + (g_1(x) z+g_2(x) z^m) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (f1[x]*y+f2[x]*y^k)*D[w[x, y,z], y] +(g1[x]*z+g2[x]*z^m)*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left ((k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\text {f1}(K[1])dK[1]\right ),(m-1) \int _1^x\exp \left ((m-1) \int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]+z^{1-m} \exp \left ((m-1) \int _1^x\text {g1}(K[3])dK[3]\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y,z),x)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+(g1(x)*z+g2(x)*z^m)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{1-k}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}, \left ( m-1 \right ) \int \!{{\rm e}^{ \left ( m-1 \right ) \int \!{\it g1} \left ( x \right ) \,{\rm d}x}}{\it g2} \left ( x \right ) \,{\rm d}x+{z}^{-m+1}{{\rm e}^{ \left ( m-1 \right ) \int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) \]

____________________________________________________________________________________

6.6.23.9 [1559] Problem 9

problem number 1559

Added May 31, 2019.

Problem Chapter 6.8.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (f_1(x) y+f_2(x) y^k) w_y + (g_1(x)+g_2(x) e^{\lambda z}) w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (f1[x]*y+f2[x]*y^k)*D[w[x, y,z], y] +(g1[x]+g2[x]*Exp[lambda*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 :=  diff(w(x,y,z),x)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+(g1(x)+g2(x)*exp(lambda*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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}}{\it f2} \left ( x \right ) \,{\rm d}x+{y}^{1-k}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it f1} \left ( x \right ) \,{\rm d}x}},{\frac {-\lambda \,\int \!{\it g2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-{{\rm e}^{\lambda \, \left ( \int \!{\it g1} \left ( x \right ) \,{\rm d}x-z \right ) }}}{\lambda }} \right ) \]

____________________________________________________________________________________

6.6.23.10 [1560] Problem 10

problem number 1560

Added May 31, 2019.

Problem Chapter 6.8.1.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (f_1(x)+f_2(x) e^{\lambda y}) w_y + (g_1(x)+g_2(x) e^{\beta z}) w_z = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (f1[x]+f2[x]*Exp[lambda*y])*D[w[x, y,z], y] +(g1[x]+g2[x]*Exp[beta*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 :=  diff(w(x,y,z),x)+ (f1(x)+f2(x)*exp(lambda*y))*diff(w(x,y,z),y)+(g1(x)+g2(x)*exp(beta*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'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-\lambda \,\int \!{\it f2} \left ( x \right ) {{\rm e}^{\lambda \,\int \!{\it f1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-{{\rm e}^{-\lambda \, \left ( y-\int \!{\it f1} \left ( x \right ) \,{\rm d}x \right ) }}}{\lambda }},{\frac {-\beta \,\int \!{\it g2} \left ( x \right ) {{\rm e}^{\beta \,\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-{{\rm e}^{\beta \, \left ( \int \!{\it g1} \left ( x \right ) \,{\rm d}x-z \right ) }}}{\beta }} \right ) \]

____________________________________________________________________________________