6.6.23 8.1

6.6.23.1 [1550] Problem 1

problem number 1550

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-\int f \left (x \right )d x +y , -\int g \left (x \right )d x +z \right )\]

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6.6.23.2 [1551] Problem 2

problem number 1551

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'));
 

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

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6.6.23.3 [1552] Problem 3

problem number 1552

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'));
 

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

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6.6.23.4 [1553] Problem 4

problem number 1553

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-\int \operatorname {f2} \left (x \right ) {\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x}d x +{\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x} y , -\int _{}^{x}\left (\operatorname {g1} \left (\textit {\_f} \right ) y \,{\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f} -\int \operatorname {f1} \left (x \right )d x}+\operatorname {g1} \left (\textit {\_f} \right ) {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}} \int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{-\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} -{\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}} \operatorname {g1} \left (\textit {\_f} \right ) \int \operatorname {f2} \left (x \right ) {\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x}d x +\operatorname {g2} \left (\textit {\_f} \right )\right )d \textit {\_f} +z \right )\]

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6.6.23.5 [1554] Problem 5

problem number 1554

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-\int \operatorname {f2} \left (x \right ) {\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x}d x +{\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x} y , {\mathrm e}^{-\int \operatorname {g1} \left (x \right )d x} z -\int \operatorname {g2} \left (x \right ) {\mathrm e}^{-\int \operatorname {g1} \left (x \right )d x}d x \right )\]

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6.6.23.6 [1555] Problem 6

problem number 1555

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=()

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6.6.23.7 [1556] Problem 7

problem number 1556

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {2 \lambda \exp \left (2 \lambda x-2 \int _1^{e^{\lambda x}}-\frac {a K[1]^2-\lambda K[1]+a}{2 \lambda K[1]^2}dK[1]\right )}{a e^{2 \lambda x}+a-2 y e^{\lambda x}}-\int _1^{e^{\lambda x}}\exp \left (-2 \int _1^{K[2]}-\frac {a K[1]^2-\lambda K[1]+a}{2 \lambda K[1]^2}dK[1]\right )dK[2],-\frac {\gamma \int _1^xf(K[3])dK[3]+\text {arctanh}(\cosh (\gamma z))}{\gamma }\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {\left (\left (-2 \cosh \left (\lambda x \right ) a -2 y \right ) \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, \frac {-8 i a +3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )+i \cosh \left (\lambda x \right ) \operatorname {HeunCPrime}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, \frac {-8 i a +3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \lambda \right ) \sqrt {\sinh \left (\lambda x \right )+i}}{\left (\left (2 i a +2 \sinh \left (\lambda x \right ) a +\lambda \right ) \cosh \left (\lambda x \right )+2 y \left (\sinh \left (\lambda x \right )+i\right )\right ) \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, \frac {-8 i a +3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )-\operatorname {HeunCPrime}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, \frac {-8 i a +3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \lambda \cosh \left (\lambda x \right ) \left (i \sinh \left (\lambda x \right )-1\right )}, \frac {-\int f \left (x \right )d x \gamma -2 \,\operatorname {arctanh}\left ({\mathrm e}^{\gamma z}\right )}{\gamma }\right )\]

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6.6.23.8 [1557] Problem 8

problem number 1557

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'));
 

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

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6.6.23.9 [1558] Problem 9

problem number 1558

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'));
 

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

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6.6.23.10 [1559] Problem 10

problem number 1559

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {-{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}-\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda }{\lambda }, \frac {-{\mathrm e}^{\beta \left (-z +\int \operatorname {g1} \left (x \right )d x \right )}-\int \operatorname {g2} \left (x \right ) {\mathrm e}^{\beta \int \operatorname {g1} \left (x \right )d x}d x \beta }{\beta }\right )\]

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