6.8.24 8.2

6.8.24.1 [1900] Problem 1

problem number 1900

Added December 1, 2019.

Problem Chapter 8.8.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + g(y) w_y + h(z) w_z = \left ( \varphi (z)+\psi (y)+\chi (z) \right ) w \]

Mathematica ✗

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

Failed

Maple ✓

restart; 
local gamma; 
pde :=  f(x)*diff(w(x,y,z),x)+ g(y)*diff(w(x,y,z),y)+ h(x)*diff(w(x,y,z),z)=(varphi(z)+psi(y)+chi(z))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-\int \frac {1}{f \left (x \right )}d x +\int \frac {1}{g \left (y \right )}d y , -\int \frac {h \left (x \right )}{f \left (x \right )}d x +z \right ) {\mathrm e}^{\int _{}^{x}\frac {\varphi \left (\int \frac {h \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g} -\int \frac {h \left (x \right )}{f \left (x \right )}d x +z \right )+\psi \left (\operatorname {RootOf}\left (\int \frac {1}{f \left (\textit {\_g} \right )}d \textit {\_g} -\int _{}^{\textit {\_Z}}\frac {1}{g \left (\textit {\_a} \right )}d \textit {\_a} -\int \frac {1}{f \left (x \right )}d x +\int \frac {1}{g \left (y \right )}d y \right )\right )+\chi \left (\int \frac {h \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g} -\int \frac {h \left (x \right )}{f \left (x \right )}d x +z \right )}{f \left (\textit {\_g} \right )}d \textit {\_g}}\]

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6.8.24.2 [1901] Problem 2

problem number 1901

Added December 1, 2019.

Problem Chapter 8.8.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + z w_y + g(y) w_z = \left ( h_2(x)+h_1(y) \right ) w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  f[x]*D[w[x,y,z],x]+z*D[w[x,y,z],y]+g[y]*D[w[x,y,z],z]==(h2[x]+h1[y])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  f(x)*diff(w(x,y,z),x)+ z*diff(w(x,y,z),y)+ g(y)*diff(w(x,y,z),z)=(h__2(x)+h__1(y))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{1} \left (z^{2}-2 \int g \left (y \right )d y , -\int _{}^{y}\frac {1}{\sqrt {2 \int g \left (\textit {\_b} \right )d \textit {\_b} +z^{2}-2 \int g \left (y \right )d y}}d \textit {\_b} +\int \frac {1}{f \left (x \right )}d x \right ) {\mathrm e}^{\int _{}^{y}\frac {h_{2} \left (\operatorname {RootOf}\left (\int \frac {1}{\sqrt {2 \int g \left (\textit {\_b} \right )d \textit {\_b} +z^{2}-2 \int g \left (y \right )d y}}d \textit {\_b} -\int _{}^{\textit {\_Z}}\frac {1}{f \left (\textit {\_b} \right )}d \textit {\_b} -\int _{}^{y}\frac {1}{\sqrt {2 \int g \left (\textit {\_b} \right )d \textit {\_b} +z^{2}-2 \int g \left (y \right )d y}}d \textit {\_b} +\int \frac {1}{f \left (x \right )}d x \right )\right )+h_{1} \left (\textit {\_b} \right )}{\sqrt {2 \int g \left (\textit {\_b} \right )d \textit {\_b} +z^{2}-2 \int g \left (y \right )d y}}d \textit {\_b}}\]

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6.8.24.3 [1902] Problem 3

problem number 1902

Added December 1, 2019.

Problem Chapter 8.8.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f_1(x) w_x + f_2(x) g(y) w_y + f_3(x) h(z) w_z = f_4(x) w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  f1[x]*D[w[x,y,z],x]+f2[x]*g[y]*D[w[x,y,z],y]+f3[x]*h[z]*D[w[x,y,z],z]==f4[x]*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  f__1(x)*diff(w(x,y,z),x)+ f__2(x)*g(y)*diff(w(x,y,z),y)+ f__3(x)*h(z)*diff(w(x,y,z),z)=f__4(x)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{5} \left (-\int \frac {f_{2} \left (x \right )}{f_{1} \left (x \right )}d x +\int \frac {1}{g \left (y \right )}d y , -\int \frac {f_{3} \left (x \right )}{f_{1} \left (x \right )}d x +\int \frac {1}{h \left (z \right )}d z \right ) {\mathrm e}^{\int \frac {f_{4} \left (x \right )}{f_{1} \left (x \right )}d x}\]

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6.8.24.4 [1903] Problem 4

problem number 1903

Added December 1, 2019.

Problem Chapter 8.8.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f_1(x) y+f_2(x) \right ) w_y + \left (g_1(x) z+g_2(y) \right ) w_z = \left ( h_1(x)+h_2(y) \right ) w \]

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]==(h1[x]+h2[y])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^x\left (\text {h1}(K[5])+\text {h2}\left (\exp \left (\int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[5]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )\right )dK[5]\right ) c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2],z \exp \left (-\int _1^x\text {g1}(K[3])dK[3]\right )-\int _1^x\exp \left (-\int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (f__1(x)*y+f__2(x))*diff(w(x,y,z),y)+ (g__1(x)*z+g__2(x))*diff(w(x,y,z),z)=(h__1(x)+h__2(y))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{3} \left ({\mathrm e}^{-\int f_{1} \left (x \right )d x} y -\int f_{2} \left (x \right ) {\mathrm e}^{-\int f_{1} \left (x \right )d x}d x , {\mathrm e}^{-\int g_{1} \left (x \right )d x} z -\int g_{2} \left (x \right ) {\mathrm e}^{-\int g_{1} \left (x \right )d x}d x \right ) {\mathrm e}^{\int _{}^{x}\left (h_{1} \left (\textit {\_f} \right )+h_{2} \left (\left (\int f_{2} \left (\textit {\_f} \right ) {\mathrm e}^{-\int f_{1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} +{\mathrm e}^{-\int f_{1} \left (x \right )d x} y -\int f_{2} \left (x \right ) {\mathrm e}^{-\int f_{1} \left (x \right )d x}d x \right ) {\mathrm e}^{\int f_{1} \left (\textit {\_f} \right )d \textit {\_f}}\right )\right )d \textit {\_f}}\]

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6.8.24.5 [1904] Problem 5

problem number 1904

Added December 1, 2019.

Problem Chapter 8.8.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f_1(x) y+f_2(x) y^k\right ) w_y + \left (g_1(y) z+g_2(x) z^m \right ) w_z = \left ( h_1(x)+h_2(y) \right ) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(f1[x]*y+f2[x]*y^k)*D[w[x,y,z],y]+(g1[y]*z+g2[x])*D[w[x,y,z],z]==(h1[x]+h2[y])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^x\left (\text {h1}(K[5])+\text {h2}\left (\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (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^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[5]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )\right )dK[5]\right ) 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 ),z \exp \left (-\int _1^x\text {g1}\left (\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (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^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[3]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )dK[3]\right )-\int _1^x\exp \left (-\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (\exp \left (\int _1^x\text {g1}\left (\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (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^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[3]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )dK[3]\right )\right ),\{K[3],1,x\}\right ]dK[3]\right ) \text {g2}(K[4])dK[4]\right )\right \}\right \}\]

Maple ✗

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (f__1(y)*y+f__2(x)*y^k)*diff(w(x,y,z),y)+ (g__1(y)*z+g__2(x))*diff(w(x,y,z),z)=(h__1(x)+h__2(y))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

sol=()

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6.8.24.6 [1905] Problem 6

problem number 1905

Added December 1, 2019.

Problem Chapter 8.8.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f_1(x) y+f_2(x) y^k\right ) w_y + \left (g_1(x)+g_2(y) e^{\lambda z} \right ) w_z = \left ( h_1(x)+h_2(y) \right ) w \]

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]==(h1[x]+h2[y])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (f__1(x)*y+f__2(x)*y^k)*diff(w(x,y,z),y)+ (g__1(x)+g__2(x)*exp(lambda*z))*diff(w(x,y,z),z)=(h__1(x)+h__2(y))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = f_{3} \left (\left (k -1\right ) \int f_{2} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int f_{1} \left (x \right )d x}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int f_{1} \left (x \right )d x}, \frac {-\int g_{2} \left (x \right ) {\mathrm e}^{\lambda \int g_{1} \left (x \right )d x}d x \lambda -{\mathrm e}^{\lambda \left (-z +\int g_{1} \left (x \right )d x \right )}}{\lambda }\right ) {\mathrm e}^{\int _{}^{x}\left (h_{2} \left (\left (\left (1-k \right ) \int f_{2} \left (\textit {\_f} \right ) {\mathrm e}^{\left (k -1\right ) \int f_{1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} +\left (k -1\right ) \int f_{2} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int f_{1} \left (x \right )d x}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int f_{1} \left (x \right )d x}\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int f_{1} \left (\textit {\_f} \right )d \textit {\_f}}\right )+h_{1} \left (\textit {\_f} \right )\right )d \textit {\_f}}\]

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6.8.24.7 [1906] Problem 7

problem number 1906

Added December 1, 2019.

Problem Chapter 8.8.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f_1(x) y+f_2(x) e^{\lambda y}\right ) w_y + \left (g_1(y) z+g_2(x) z^k \right ) w_z = \left ( h_1(x)+h_2(y) \right ) w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(f1[x]*y+f2[x]*Exp[lambda*y])*D[w[x,y,z],y]+(g1[y]*z+g2[x]*z^k)*D[w[x,y,z],z]==(h1[x]+h2[y])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✗

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (f__1(x)*y+f__2(x)*exp(lambda*y))*diff(w(x,y,z),y)+ (g__1(y)*z+g__2(x)*z^k)*diff(w(x,y,z),z)=(h__1(x)+h__2(y))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

sol=()

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6.8.24.8 [1907] Problem 8

problem number 1907

Added December 1, 2019.

Problem Chapter 8.8.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f_1(x) y+f_2(x) e^{\lambda y}\right ) w_y + \left (g_1(x)+g_2(x) e^{\beta z} \right ) w_z = \left ( h_1(x)+h_2(y) \right ) w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(f1[x]*y+f2[x]*Exp[lambda*y])*D[w[x,y,z],y]+(g1[x]+g2[y]*Exp[beta*z])*D[w[x,y,z],z]==(h1[x]+h2[y])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✗

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (f__1(x)*y+f__2(x)*exp(lambda*y))*diff(w(x,y,z),y)+ (g__1(x)+g__2(y)*exp(beta*z))*diff(w(x,y,z),z)=(h__1(x)+h__2(y))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

sol=()

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