6.8.23 8.1

6.8.23.1 [1891] Problem 1

problem number 1891

Added December 1, 2019.

Problem Chapter 8.8.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + f(x) w_y + g(x) w_z = \left ( h_2(x) y+h_1(x) z+h_0(x) \right ) w \]

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]==(h2[x]*y+h1[x]*z+h0[x])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\int _1^xf(K[1])dK[1],z-\int _1^xg(K[2])dK[2]\right ) \exp \left (\int _1^x\left (\text {h0}(K[3])+\text {h2}(K[3]) \left (y-\int _1^xf(K[1])dK[1]+\int _1^{K[3]}f(K[1])dK[1]\right )+\text {h1}(K[3]) \left (z-\int _1^xg(K[2])dK[2]+\int _1^{K[3]}g(K[2])dK[2]\right )\right )dK[3]\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ f(x)*diff(w(x,y,z),y)+ g(x)*diff(w(x,y,z),z)= (h2(x)*y+h1(x)*z+h0(x))*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y -\left (\int f \left (x \right )d x \right ), z -\left (\int g \left (x \right )d x \right )\right ) {\mathrm e}^{\int _{}^{x}\left (\left (\int f \left (\mathit {\_f} \right )d \mathit {\_f} \right ) \mathit {h2} \left (\mathit {\_f} \right )+\left (\int g \left (\mathit {\_f} \right )d \mathit {\_f} \right ) \mathit {h1} \left (\mathit {\_f} \right )+\mathit {h0} \left (\mathit {\_f} \right )+\left (z -\left (\int g \left (x \right )d x \right )\right ) \mathit {h1} \left (\mathit {\_f} \right )+\left (y -\left (\int f \left (x \right )d x \right )\right ) \mathit {h2} \left (\mathit {\_f} \right )\right )d\mathit {\_f}}\]

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6.8.23.2 [1892] Problem 2

problem number 1892

Added December 1, 2019.

Problem Chapter 8.8.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + f(x)(y+a) w_y + g(x)(z+b) w_z = h(x) w \]

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

\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^xh(K[5])dK[5]\right ) c_1\left (y \exp \left (-\int _1^xf(K[1])dK[1]\right )-\int _1^xa \exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right ) f(K[2])dK[2],z \exp \left (-\int _1^xg(K[3])dK[3]\right )-\int _1^xb \exp \left (-\int _1^{K[4]}g(K[3])dK[3]\right ) g(K[4])dK[4]\right )\right \}\right \}\]

Maple ✓

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

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

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6.8.23.3 [1893] Problem 3

problem number 1893

Added December 1, 2019.

Problem Chapter 8.8.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + \left ( a y+f(x) \right ) w_y + \left ( b z+g(x) \right ) w_z = h(x) w \]

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

\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^xh(K[3])dK[3]\right ) c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} f(K[1])dK[1],z e^{-b x}-\int _1^xe^{-b K[2]} g(K[2])dK[2]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y \,{\mathrm e}^{-a x}-\left (\int {\mathrm e}^{-a x} f \left (x \right )d x \right ), z \,{\mathrm e}^{-b x}-\left (\int {\mathrm e}^{-b x} g \left (x \right )d x \right )\right ) {\mathrm e}^{\int h \left (x \right )d x}\]

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6.8.23.4 [1894] Problem 4

problem number 1894

Added December 1, 2019.

Problem Chapter 8.8.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + \left ( f_1(x) y+ f_2(x) \right ) w_y + \left ( g_1(x) y+ g_2(x) \right ) w_z = \left ( h_2(x) y+h_1(x) z+h_0(x) \right ) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+(f1[x]*y+f2[x])*D[w[x,y,z],y]+(g1[x]*y+g2[x])*D[w[x,y,z],z]==(h2[x]*y+h1[x]*z+h0[x])*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 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 ) \exp \left (\int _1^x\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \text {h0}(K[5])+\exp \left (\int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) \text {h2}(K[5]) \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]+\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \int _1^{K[5]}\exp \left (-\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3]\right )+\text {h1}(K[5]) \left (-y \int _1^x\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]+\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]+\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \int _1^{K[5]}\exp \left (-\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3]\right ) \int _1^{K[5]}\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]+\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \left (z-\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]+\int _1^{K[5]}\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]\right )\right )\right )dK[5]\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
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)= (h2(x)*y+h1(x)*z+h0(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 ) = \mathit {\_F1} \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), z -\left (\int _{}^{x}\left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )+\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )+\mathit {g2} \left (\mathit {\_f} \right )\right )d\mathit {\_f} \right )\right ) {\mathrm e}^{\int _{}^{x}\left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}} \mathit {h2} \left (\mathit {\_g} \right )+\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {f2} \left (\mathit {\_g} \right )d \mathit {\_g} \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}} \mathit {h2} \left (\mathit {\_g} \right )-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}} \mathit {h2} \left (\mathit {\_g} \right )+z \mathit {h1} \left (\mathit {\_g} \right )+\left (\int \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}} \mathit {g1} \left (\mathit {\_g} \right )+\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {f2} \left (\mathit {\_g} \right )d \mathit {\_g} \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}} \mathit {g1} \left (\mathit {\_g} \right )-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}} \mathit {g1} \left (\mathit {\_g} \right )+\mathit {g2} \left (\mathit {\_g} \right )\right )d \mathit {\_g} \right ) \mathit {h1} \left (\mathit {\_g} \right )-\left (\int _{}^{x}\left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )+\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )+\mathit {g2} \left (\mathit {\_f} \right )\right )d\mathit {\_f} \right ) \mathit {h1} \left (\mathit {\_g} \right )+\mathit {h0} \left (\mathit {\_g} \right )\right )d\mathit {\_g}}\]

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6.8.23.5 [1895] Problem 5

problem number 1895

Added December 1, 2019.

Problem Chapter 8.8.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + \left ( f_1(x) y+ f_2(x) \right ) w_y + \left ( g_1(x) z+ g_2(x) \right ) w_z = \left ( h_2(x) y+h_1(x) z+h_0(x) \right ) w \]

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]==(h2[x]*y+h1[x]*z+h0[x])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2],z \exp \left (-\int _1^x\text {g1}(K[3])dK[3]\right )-\int _1^x\exp \left (-\int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]\right ) \exp \left (\int _1^x\left (\text {h0}(K[5])+\exp \left (\int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) \text {h2}(K[5]) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[5]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )+\exp \left (\int _1^{K[5]}\text {g1}(K[3])dK[3]\right ) \text {h1}(K[5]) \left (\exp \left (-\int _1^x\text {g1}(K[3])dK[3]\right ) z-\int _1^x\exp \left (-\int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]+\int _1^{K[5]}\exp \left (-\int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]\right )\right )dK[5]\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
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)= (h2(x)*y+h1(x)*z+h0(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 ) = \mathit {\_F1} \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), z \,{\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {g2} \left (x \right )d x \right )\right ) {\mathrm e}^{\int _{}^{x}\left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}} \mathit {h2} \left (\mathit {\_h} \right )+z \,{\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )} {\mathrm e}^{\int \mathit {g1} \left (\mathit {\_h} \right )d \mathit {\_h}} \mathit {h1} \left (\mathit {\_h} \right )+\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {f2} \left (\mathit {\_h} \right )d \mathit {\_h} \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}} \mathit {h2} \left (\mathit {\_h} \right )-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_h} \right )d \mathit {\_h}} \mathit {h2} \left (\mathit {\_h} \right )+\left (\int {\mathrm e}^{-\left (\int \mathit {g1} \left (\mathit {\_h} \right )d \mathit {\_h} \right )} \mathit {g2} \left (\mathit {\_h} \right )d \mathit {\_h} \right ) {\mathrm e}^{\int \mathit {g1} \left (\mathit {\_h} \right )d \mathit {\_h}} \mathit {h1} \left (\mathit {\_h} \right )-\left (\int {\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {g2} \left (x \right )d x \right ) {\mathrm e}^{\int \mathit {g1} \left (\mathit {\_h} \right )d \mathit {\_h}} \mathit {h1} \left (\mathit {\_h} \right )+\mathit {h0} \left (\mathit {\_h} \right )\right )d\mathit {\_h}}\]

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6.8.23.6 [1896] Problem 6

problem number 1896

Added December 1, 2019.

Problem Chapter 8.8.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + \left ( y^2-a^2+a \lambda \sinh (\lambda x)-a^2 \sinh ^2(\lambda x) \right ) w_y + f(x) \sinh (\gamma z) w_z = g(x) w \]

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-\frac {2 \sqrt {\sinh \left (\lambda x \right )+i}\, \left (-\left (-\frac {\left (\sinh ^{2}\left (\lambda x \right )\right )}{2}+i \sinh \left (\lambda x \right )+\frac {1}{2}\right ) \lambda \HeunCPrime \left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cosh \left (\lambda x \right )+\left (i \left (\sinh ^{2}\left (\lambda x \right )\right )+2 \sinh \left (\lambda x \right )-i\right ) \left (a \cosh \left (\lambda x \right )+y \right ) \HeunC \left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )\right )}{-\left (\sinh ^{2}\left (\lambda x \right )+1\right ) \left (-\sinh \left (\lambda x \right )+i\right ) \lambda \HeunCPrime \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cosh \left (\lambda x \right )+\left (2 \left (\sinh ^{2}\left (\lambda x \right )+1\right ) \left (i \sinh \left (\lambda x \right )+1\right ) y +\left (2 i a \left (\sinh ^{3}\left (\lambda x \right )\right )+\left (i \lambda +2 a \right ) \left (\sinh ^{2}\left (\lambda x \right )\right )+2 a -i \lambda +\left (2 i a +2 \lambda \right ) \sinh \left (\lambda x \right )\right ) \cosh \left (\lambda x \right )\right ) \HeunC \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )}, \frac {-\gamma \left (\int f \left (x \right )d x \right )-2 \arctanh \left ({\mathrm e}^{\gamma z}\right )}{\gamma }\right ) {\mathrm e}^{\int g \left (x \right )d x}\]

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6.8.23.7 [1897] Problem 7

problem number 1897

Added December 1, 2019.

Problem Chapter 8.8.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + \left ( f_1(x) y+f_2(x) y^k \right ) w_y + \left ( g_1(x) z+g_2(x) z^m \right ) w_z = h(x) w \]

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]*z+g2[x]*z^m)*D[w[x,y,z],z]==h[x]*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

Maple ✓

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

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

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6.8.23.8 [1898] Problem 8

problem number 1898

Added December 1, 2019.

Problem Chapter 8.8.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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6.8.23.9 [1899] Problem 9

problem number 1899

Added December 1, 2019.

Problem Chapter 8.8.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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