7.6.25 8.3

7.6.25.1 [1569] Problem 1

problem number 1569

Added May 31, 2019.

Problem Chapter 6.8.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$a{w}_x+b{w}_y+f(x,y)w_z=0$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to c_1(y-\frac{bx}{a},z-{\int }_1^x\frac{f(K[1],y+\frac{b(K[1]-x)}{a})}{a}dK[1])\}\right\}$$

Maple ✓

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

$$w(x,y,z)=\mathit{\text{\_F1}}(\frac{ay-bx}{a},z-\left({\int }_{}^x\frac{f(\mathit{\text{\_a}},\frac{ay-(-\mathit{\text{\_a}}+x)b}{a})}{a}d\mathit{\text{\_a}}\right))$$

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7.6.25.2 [1570] Problem 2

problem number 1570

Added May 31, 2019.

Problem Chapter 6.8.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$a{w}_x+b{w}_y+f(x,y)g(z)w_z=0$$

Mathematica ✓

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

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

Maple ✓

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

$$w(x,y,z)=\mathit{\text{\_F1}}(\frac{ay-bx}{a},\int \frac{a}{g\left(z\right)}dz-\left({\int }_{}^{x}f(\mathit{\text{\_a}},\frac{ay-(-\mathit{\text{\_a}}+x)b}{a})d\mathit{\text{\_a}}\right))$$

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7.6.25.3 [1571] Problem 3

problem number 1571

Added May 31, 2019.

Problem Chapter 6.8.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$x{w}_x+y{w}_y+(z+f(x,y))w_z=0$$

Mathematica ✓

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

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

Maple ✓

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

$$w(x,y,z)=\mathit{\text{\_F1}}(\frac{y}{x},\frac{-x\left({\int }_{}^x\frac{f(\mathit{\text{\_a}},\frac{\mathit{\text{\_a}}y}{x})}{{\mathit{\text{\_a}}}^2}d\mathit{\text{\_a}}\right)+z}{x})$$

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7.6.25.4 [1572] Problem 4

problem number 1572

Added May 31, 2019.

Problem Chapter 6.8.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$ax{w}_x+by{w}_y+f(x,y)w_z=0$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to c_1(y{x}^{-\frac{b}{a}},z-{\int }_1^x\frac{f(K[1],x^{-\frac{b}{a}}yK[1{]}^{\frac{b}{a}})}{aK[1]}dK[1])\}\right\}$$

Maple ✓

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

$$w(x,y,z)=\mathit{\text{\_F1}}(y{x}^{-\frac{b}{a}},z-\left({\int }_{}^x\frac{f(\mathit{\text{\_a}},y{\mathit{\text{\_a}}}^{\frac{b}{a}}{x}^{-\frac{b}{a}})}{\mathit{\text{\_a}}a}d\mathit{\text{\_a}}\right))$$

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7.6.25.5 [1573] Problem 5

problem number 1573

Added May 31, 2019.

Problem Chapter 6.8.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$ax{w}_x+by{w}_y+f(x,y)g(z)w_z=0$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to c_1(y{x}^{-\frac{b}{a}},z-{\int }_1^x\frac{f(K[1],x^{-\frac{b}{a}}yK[1{]}^{\frac{b}{a}})g(K[1])}{aK[1]}dK[1])\}\right\}$$

Maple ✓

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

$$w(x,y,z)=\mathit{\text{\_F1}}(y{x}^{-\frac{b}{a}},\int \frac{a}{g\left(z\right)}dz-\left({\int }_{}^x\frac{f(\mathit{\text{\_a}},y{\mathit{\text{\_a}}}^{\frac{b}{a}}{x}^{-\frac{b}{a}})}{\mathit{\text{\_a}}}d\mathit{\text{\_a}}\right))$$

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7.6.25.6 [1574] Problem 6

problem number 1574

Added May 31, 2019.

Problem Chapter 6.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$w_x+(f_1(x)y+f_2(x))w_y+(g(x,y)z+h(x,y))w_z=0$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to c_1(y\exp (-{\int }_1^x\text{f1}(K[1])dK[1])-{\int }_1^x\exp (-{\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2],z\exp (-{\int }_1^{x}g(K[3],\exp ({\int }_1^{K[3]}\text{f1}(K[1])dK[1])(\exp (-{\int }_1^x\text{f1}(K[1])dK[1])y-{\int }_1^x\exp (-{\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2]+{\int }_1^{K[3]}\exp (-{\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2]))dK[3])-{\int }_1^x\exp (-{\int }_1^{K[4]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][\log (\exp ({\int }_1^{x}g(K[3],\exp ({\int }_1^{K[3]}\text{f1}(K[1])dK[1])(\exp (-{\int }_1^x\text{f1}(K[1])dK[1])y-{\int }_1^x\exp (-{\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2]+{\int }_1^{K[3]}\exp (-{\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2]))dK[3])),\{K[3],1,x\}]dK[3])h(K[4],\exp ({\int }_1^{K[4]}\text{f1}(K[1])dK[1])(\exp (-{\int }_1^x\text{f1}(K[1])dK[1])y-{\int }_1^x\exp (-{\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2]+{\int }_1^{K[4]}\exp (-{\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2]))dK[4])\}\right\}$$

Maple ✓

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

$$w(x,y,z)=\mathit{\text{\_F1}}(y{\mathrm{e}}^{-(\int \mathit{f}\mathit{1}\left(x\right)dx)}-(\int {\mathrm{e}}^{-(\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx),z{\mathrm{e}}^{-\left({\int }_{}^{x}g(\mathit{\text{\_f}},(y{\mathrm{e}}^{-(\int \mathit{f}\mathit{1}\left(x\right)dx)}+\int {\mathrm{e}}^{-(\int \mathit{f}\mathit{1}\left(\mathit{\text{\_f}}\right)d\mathit{\text{\_f}})}\mathit{f}\mathit{2}\left(\mathit{\text{\_f}}\right)d\mathit{\text{\_f}}-(\int {\mathrm{e}}^{-(\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)){\mathrm{e}}^{\int \mathit{f}\mathit{1}\left(\mathit{\text{\_f}}\right)d\mathit{\text{\_f}}})d\mathit{\text{\_f}}\right)}-\left({\int }_{}^x{\mathrm{e}}^{-(\int g(\mathit{\text{\_a}},(y{\mathrm{e}}^{-(\int \mathit{f}\mathit{1}\left(x\right)dx)}+\int {\mathrm{e}}^{-(\int \mathit{f}\mathit{1}\left(\mathit{\text{\_a}}\right)d\mathit{\text{\_a}})}\mathit{f}\mathit{2}\left(\mathit{\text{\_a}}\right)d\mathit{\text{\_a}}-(\int {\mathrm{e}}^{-(\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)){\mathrm{e}}^{\int \mathit{f}\mathit{1}\left(\mathit{\text{\_a}}\right)d\mathit{\text{\_a}}})d\mathit{\text{\_a}})}h(\mathit{\text{\_a}},(y{\mathrm{e}}^{-(\int \mathit{f}\mathit{1}\left(x\right)dx)}+\int {\mathrm{e}}^{-(\int \mathit{f}\mathit{1}\left(\mathit{\text{\_a}}\right)d\mathit{\text{\_a}})}\mathit{f}\mathit{2}\left(\mathit{\text{\_a}}\right)d\mathit{\text{\_a}}-(\int {\mathrm{e}}^{-(\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)){\mathrm{e}}^{\int \mathit{f}\mathit{1}\left(\mathit{\text{\_a}}\right)d\mathit{\text{\_a}}})d\mathit{\text{\_a}}\right))$$

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7.6.25.7 [1575] Problem 7

problem number 1575

Added May 31, 2019.

Problem Chapter 6.8.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$w_x+(f_1(x)y+f_2(x)y^k)w_y+(g(x,y)z+h(x,y)z^m)w_z=0$$

Mathematica ✓

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

$$\left\{\{w(x,y,z)\to c_1((k-1){\int }_1^x\exp ((k-1){\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2]+y^{1-k}\exp ((k-1){\int }_1^x\text{f1}(K[1])dK[1]),(m-1){\int }_1^x\exp ((m-1){\int }_1^{K[4]}\text{InverseFunction}[\text{Inactive}[\text{Integrate}],1,2][-\frac{\log (\exp (-((m-1){\int }_1^{x}g(K[3],(\exp (-{\int }_1^x\text{f1}(K[1])dK[1]-(k-1){\int }_1^{K[3]}\text{f1}(K[1])dK[1])y^{-k}(\exp ({\int }_1^x\text{f1}(K[1])dK[1])(k-1){\int }_1^x\exp ((k-1){\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2]y^k-\exp ({\int }_1^x\text{f1}(K[1])dK[1])(k-1){\int }_1^{K[3]}\exp ((k-1){\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2]y^k+\exp (k{\int }_1^x\text{f1}(K[1])dK[1])y)){}^{\frac{1}{1-k}})dK[3])))}{m-1},\{K[3],1,x\}]dK[3])h(K[4],(\exp (-{\int }_1^x\text{f1}(K[1])dK[1]-(k-1){\int }_1^{K[4]}\text{f1}(K[1])dK[1])y^{-k}(\exp ({\int }_1^x\text{f1}(K[1])dK[1])(k-1){\int }_1^x\exp ((k-1){\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2]y^k-\exp ({\int }_1^x\text{f1}(K[1])dK[1])(k-1){\int }_1^{K[4]}\exp ((k-1){\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2]y^k+\exp (k{\int }_1^x\text{f1}(K[1])dK[1])y)){}^{\frac{1}{1-k}})dK[4]+z^{1-m}\exp ((m-1){\int }_1^{x}g(K[3],(\exp (-{\int }_1^x\text{f1}(K[1])dK[1]-(k-1){\int }_1^{K[3]}\text{f1}(K[1])dK[1])y^{-k}(\exp ({\int }_1^x\text{f1}(K[1])dK[1])(k-1){\int }_1^x\exp ((k-1){\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2]y^k-\exp ({\int }_1^x\text{f1}(K[1])dK[1])(k-1){\int }_1^{K[3]}\exp ((k-1){\int }_1^{K[2]}\text{f1}(K[1])dK[1])\text{f2}(K[2])dK[2]y^k+\exp (k{\int }_1^x\text{f1}(K[1])dK[1])y)){}^{\frac{1}{1-k}})dK[3]))\}\right\}$$

Maple ✓

restart; 
pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+(g(x,y)*z+h(x,y)*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'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(y^{-k+1}{\mathrm{e}}^{(k-1)(\int \mathit{f}\mathit{1}\left(x\right)dx)}+(k-1)(\int {\mathrm{e}}^{(k-1)(\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx),z^{-m+1}{\mathrm{e}}^{(m-1)\left({\int }_{}^{x}g(\mathit{\text{\_f}},{(y^{-k+1}{\mathrm{e}}^{(k-1)(\int \mathit{f}\mathit{1}\left(x\right)dx)}+(k-1)(\int {\mathrm{e}}^{(k-1)(\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)+(-k+1)(\int {\mathrm{e}}^{(k-1)(\int \mathit{f}\mathit{1}\left(\mathit{\text{\_f}}\right)d\mathit{\text{\_f}})}\mathit{f}\mathit{2}\left(\mathit{\text{\_f}}\right)d\mathit{\text{\_f}}))}^{-\frac{1}{k-1}}{\mathrm{e}}^{\int \mathit{f}\mathit{1}\left(\mathit{\text{\_f}}\right)d\mathit{\text{\_f}}})d\mathit{\text{\_f}}\right)}+(m-1)\left({\int }_{}^x{\mathrm{e}}^{(m-1)(\int g(\mathit{\text{\_h}},{(y^{-k+1}{\mathrm{e}}^{(k-1)(\int \mathit{f}\mathit{1}\left(x\right)dx)}+(k-1)(\int {\mathrm{e}}^{(k-1)(\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)+(-k+1)(\int {\mathrm{e}}^{(k-1)(\int \mathit{f}\mathit{1}\left(\mathit{\text{\_h}}\right)d\mathit{\text{\_h}})}\mathit{f}\mathit{2}\left(\mathit{\text{\_h}}\right)d\mathit{\text{\_h}}))}^{-\frac{1}{k-1}}{\mathrm{e}}^{\int \mathit{f}\mathit{1}\left(\mathit{\text{\_h}}\right)d\mathit{\text{\_h}}})d\mathit{\text{\_h}})}h(\mathit{\text{\_h}},{(y^{-k+1}{\mathrm{e}}^{(k-1)(\int \mathit{f}\mathit{1}\left(x\right)dx)}+(k-1)(\int {\mathrm{e}}^{(k-1)(\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)+(-k+1)(\int {\mathrm{e}}^{(k-1)(\int \mathit{f}\mathit{1}\left(\mathit{\text{\_g}}\right)d\mathit{\text{\_g}})}\mathit{f}\mathit{2}\left(\mathit{\text{\_g}}\right)d\mathit{\text{\_g}}))}^{-\frac{1}{k-1}}{\mathrm{e}}^{\int \mathit{f}\mathit{1}\left(\mathit{\text{\_h}}\right)d\mathit{\text{\_h}}})d\mathit{\text{\_h}}\right))$$

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7.6.25.8 [1576] Problem 8

problem number 1576

Added May 31, 2019.

Problem Chapter 6.8.3.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$w_x+(f_1(x)y+f_2(x)y^k)w_y+(g(x,y)+h(x,y)e^{\lambda z})w_z=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (f1[x]*y+f2[x]*y^k)*D[w[x, y,z], y] +(g[x,y]*z+h[x,y]*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)+(g(x,y)*z+h(x,y)*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'));
 

time expired

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7.6.25.9 [1577] Problem 9

problem number 1577

Added May 31, 2019.

Problem Chapter 6.8.3.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$w_x+(f_1(x)+f_2(x)e^{\lambda y})w_y+(g(x,y)z+h(x,y)z^k)w_z=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (f1[x]+f2[x]*Exp[lambda*y])*D[w[x, y,z], y] +(g[x,y]*z+h[x,y]*z^k)*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)+(g(x,y)*z+h(x,y)*z^k)*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(\frac{-\lambda (\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)-{\mathrm{e}}^{-(y-(\int \mathit{f}\mathit{1}\left(x\right)dx))\lambda }}{\lambda },z^{-k+1}{\mathrm{e}}^{(k-1)\left({\int }_{}^{x}g(\mathit{\text{\_a}},\frac{\lambda (\int \mathit{f}\mathit{1}\left(\mathit{\text{\_a}}\right)d\mathit{\text{\_a}})+\ln \left(\frac{1}{(-(\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(\mathit{\text{\_a}}\right)d\mathit{\text{\_a}})}\mathit{f}\mathit{2}\left(\mathit{\text{\_a}}\right)d\mathit{\text{\_a}})+\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)\lambda +{\mathrm{e}}^{-(y-(\int \mathit{f}\mathit{1}\left(x\right)dx))\lambda }}\right)}{\lambda })d\mathit{\text{\_a}}\right)}+(k-1)\left({\int }_{}^x{\mathrm{e}}^{(k-1)(\int g(\mathit{\text{\_f}},\frac{\lambda (\int \mathit{f}\mathit{1}\left(\mathit{\text{\_f}}\right)d\mathit{\text{\_f}})+\ln \left(\frac{1}{(-(\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(\mathit{\text{\_f}}\right)d\mathit{\text{\_f}})}\mathit{f}\mathit{2}\left(\mathit{\text{\_f}}\right)d\mathit{\text{\_f}})+\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)\lambda +{\mathrm{e}}^{-(y-(\int \mathit{f}\mathit{1}\left(x\right)dx))\lambda }}\right)}{\lambda })d\mathit{\text{\_f}})}h(\mathit{\text{\_f}},\frac{\lambda (\int \mathit{f}\mathit{1}\left(\mathit{\text{\_f}}\right)d\mathit{\text{\_f}})+\ln \left(\frac{1}{(-(\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(\mathit{\text{\_f}}\right)d\mathit{\text{\_f}})}\mathit{f}\mathit{2}\left(\mathit{\text{\_f}}\right)d\mathit{\text{\_f}})+\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)\lambda +{\mathrm{e}}^{-(y-(\int \mathit{f}\mathit{1}\left(x\right)dx))\lambda }}\right)}{\lambda })d\mathit{\text{\_f}}\right))$$

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7.6.25.10 [1578] Problem 10

problem number 1578

Added May 31, 2019.

Problem Chapter 6.8.3.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$w_x+(f_1(x)+f_2(x)e^{\lambda y})w_y+(g(x,y)+h(x,y)e^{\beta z})w_z=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (f1[x]+f2[x]*Exp[lambda*y])*D[w[x, y,z], y] +(g[x,y]+h[x,y]*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)+(g(x,y)+h(x,y)*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'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(\frac{-\lambda (\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)-{\mathrm{e}}^{-(y-(\int \mathit{f}\mathit{1}\left(x\right)dx))\lambda }}{\lambda },\frac{-\beta \left({\int }_{}^x{\mathrm{e}}^{\beta (\int g(\mathit{\text{\_g}},\frac{\lambda (\int \mathit{f}\mathit{1}\left(\mathit{\text{\_g}}\right)d\mathit{\text{\_g}})+\ln \left(\frac{1}{(-(\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(\mathit{\text{\_g}}\right)d\mathit{\text{\_g}})}\mathit{f}\mathit{2}\left(\mathit{\text{\_g}}\right)d\mathit{\text{\_g}})+\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)\lambda +{\mathrm{e}}^{-(y-(\int \mathit{f}\mathit{1}\left(x\right)dx))\lambda }}\right)}{\lambda })d\mathit{\text{\_g}})}h(\mathit{\text{\_g}},\frac{\lambda (\int \mathit{f}\mathit{1}\left(\mathit{\text{\_g}}\right)d\mathit{\text{\_g}})+\ln \left(\frac{1}{(-(\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(\mathit{\text{\_g}}\right)d\mathit{\text{\_g}})}\mathit{f}\mathit{2}\left(\mathit{\text{\_g}}\right)d\mathit{\text{\_g}})+\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)\lambda +{\mathrm{e}}^{-(y-(\int \mathit{f}\mathit{1}\left(x\right)dx))\lambda }}\right)}{\lambda })d\mathit{\text{\_g}}\right)-{\mathrm{e}}^{(-z+{\int }_{}^{x}g(\mathit{\text{\_a}},\frac{\lambda (\int \mathit{f}\mathit{1}\left(\mathit{\text{\_a}}\right)d\mathit{\text{\_a}})+\ln \left(\frac{1}{(-(\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(\mathit{\text{\_a}}\right)d\mathit{\text{\_a}})}\mathit{f}\mathit{2}\left(\mathit{\text{\_a}}\right)d\mathit{\text{\_a}})+\int {\mathrm{e}}^{\lambda (\int \mathit{f}\mathit{1}\left(x\right)dx)}\mathit{f}\mathit{2}\left(x\right)dx)\lambda +{\mathrm{e}}^{-(y-(\int \mathit{f}\mathit{1}\left(x\right)dx))\lambda }}\right)}{\lambda })d\mathit{\text{\_a}})\beta }}{\beta })$$

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7.6.25.11 [1579] Problem 11

problem number 1579

Added May 31, 2019.

Problem Chapter 6.8.3.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$f_1(x)g_1(y)w_x+f_2(x)g_2(y)w_y+(h_1(x,y)+h_2(x,y)z^m)w_z=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  f1[x]*g1[y]*D[w[x, y,z], x] + f2[x]*g2[y]*D[w[x, y,z], y] +(h1[x,y]+h2[x,y]*z^m)*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 := f1(x)*g1(y)*diff(w(x,y,z),x)+ f2(x)*g2(y)*diff(w(x,y,z),y)+(h1(x,y)+h2(x,y)*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'));
 

sol=()

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7.6.25.12 [1580] Problem 12

problem number 1580

Added May 31, 2019.

Problem Chapter 6.8.3.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

$$f_1(x)g_1(y)w_x+f_2(x)g_2(y)w_y+(h_1(x,y)+h_2(x,y)e^{\lambda z})w_z=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  f1[x]*g1[y]*D[w[x, y,z], x] + f2[x]*g2[y]*D[w[x, y,z], y] +(h1[x,y]+h2[x,y]*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 := f1(x)*g1(y)*diff(w(x,y,z),x)+ f2(x)*g2(y)*diff(w(x,y,z),y)+(h1(x,y)+h2(x,y)*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'));
 

sol=()