7.8.23 8.1

7.8.23.1 [1893] Problem 1
7.8.23.2 [1894] Problem 2
7.8.23.3 [1895] Problem 3
7.8.23.4 [1896] Problem 4
7.8.23.5 [1897] Problem 5
7.8.23.6 [1898] Problem 6
7.8.23.7 [1899] Problem 7
7.8.23.8 [1900] Problem 8
7.8.23.9 [1901] Problem 9

7.8.23.1 [1893] Problem 1

problem number 1893

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) wx+f(x)wy+g(x)wz=(h2(x)y+h1(x)z+h0(x))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]];
 

{{w(x,y,z)c1(y1xf(K[1])dK[1],z1xg(K[2])dK[2])exp(1x(h0(K[3])+h2(K[3])(y1xf(K[1])dK[1]+1K[3]f(K[1])dK[1])+h1(K[3])(z1xg(K[2])dK[2]+1K[3]g(K[2])dK[2]))dK[3])}}

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(x,y,z)=_F1(y(f(x)dx),z(g(x)dx))ex((f(_f)d_f)h2(_f)+(g(_f)d_f)h1(_f)+h0(_f)+(z(g(x)dx))h1(_f)+(y(f(x)dx))h2(_f))d_f

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7.8.23.2 [1894] Problem 2

problem number 1894

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) wx+f(x)(y+a)wy+g(x)(z+b)wz=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]];
 

{{w(x,y,z)exp(1xh(K[5])dK[5])c1(yexp(1xf(K[1])dK[1])1xaexp(1K[2]f(K[1])dK[1])f(K[2])dK[2],zexp(1xg(K[3])dK[3])1xbexp(1K[4]g(K[3])dK[3])g(K[4])dK[4])}}

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(x,y,z)=_F1((a+y)e(f(x)dx),(b+z)e(g(x)dx))eh(x)dx

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7.8.23.3 [1895] Problem 3

problem number 1895

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) wx+(ay+f(x))wy+(bz+g(x))wz=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]];
 

{{w(x,y,z)exp(1xh(K[3])dK[3])c1(yeax1xeaK[1]f(K[1])dK[1],zebx1xebK[2]g(K[2])dK[2])}}

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(x,y,z)=_F1(yeax(eaxf(x)dx),zebx(ebxg(x)dx))eh(x)dx

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7.8.23.4 [1896] Problem 4

problem number 1896

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) wx+(f1(x)y+f2(x))wy+(g1(x)y+g2(x))wz=(h2(x)y+h1(x)z+h0(x))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]];
 

{{w(x,y,z)c1(yexp(1xf1(K[1])dK[1])1xexp(1K[3]f1(K[1])dK[1])f2(K[3])dK[3],1x(g2(K[4])exp(1K[4]f1(K[1])dK[1])f2(K[4])1K[4]exp(1K[2]f1(K[1])dK[1])g1(K[2])dK[2])dK[4]yexp(1xf1(K[1])dK[1])1xexp(1K[2]f1(K[1])dK[1])g1(K[2])dK[2]+z)exp(1xexp(1xf1(K[1])dK[1])(exp(1xf1(K[1])dK[1])h0(K[5])+exp(1K[5]f1(K[1])dK[1])h2(K[5])(yexp(1xf1(K[1])dK[1])1xexp(1K[3]f1(K[1])dK[1])f2(K[3])dK[3]+exp(1xf1(K[1])dK[1])1K[5]exp(1K[3]f1(K[1])dK[1])f2(K[3])dK[3])+h1(K[5])(y1xexp(1K[2]f1(K[1])dK[1])g1(K[2])dK[2]+(yexp(1xf1(K[1])dK[1])1xexp(1K[3]f1(K[1])dK[1])f2(K[3])dK[3]+exp(1xf1(K[1])dK[1])1K[5]exp(1K[3]f1(K[1])dK[1])f2(K[3])dK[3])1K[5]exp(1K[2]f1(K[1])dK[1])g1(K[2])dK[2]+exp(1xf1(K[1])dK[1])(z1x(g2(K[4])exp(1K[4]f1(K[1])dK[1])f2(K[4])1K[4]exp(1K[2]f1(K[1])dK[1])g1(K[2])dK[2])dK[4]+1K[5](g2(K[4])exp(1K[4]f1(K[1])dK[1])f2(K[4])1K[4]exp(1K[2]f1(K[1])dK[1])g1(K[2])dK[2])dK[4])))dK[5])}}

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(x,y,z)=_F1(ye(f1(x)dx)(e(f1(x)dx)f2(x)dx),z(x(ye(f1(x)dx)ef1(_f)d_fg1(_f)+(e(f1(_f)d_f)f2(_f)d_f)ef1(_f)d_fg1(_f)(e(f1(x)dx)f2(x)dx)ef1(_f)d_fg1(_f)+g2(_f))d_f))ex(ye(f1(x)dx)ef1(_g)d_gh2(_g)+(e(f1(_g)d_g)f2(_g)d_g)ef1(_g)d_gh2(_g)(e(f1(x)dx)f2(x)dx)ef1(_g)d_gh2(_g)+zh1(_g)+((ye(f1(x)dx)ef1(_g)d_gg1(_g)+(e(f1(_g)d_g)f2(_g)d_g)ef1(_g)d_gg1(_g)(e(f1(x)dx)f2(x)dx)ef1(_g)d_gg1(_g)+g2(_g))d_g)h1(_g)(x(ye(f1(x)dx)ef1(_f)d_fg1(_f)+(e(f1(_f)d_f)f2(_f)d_f)ef1(_f)d_fg1(_f)(e(f1(x)dx)f2(x)dx)ef1(_f)d_fg1(_f)+g2(_f))d_f)h1(_g)+h0(_g))d_g

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7.8.23.5 [1897] Problem 5

problem number 1897

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) wx+(f1(x)y+f2(x))wy+(g1(x)z+g2(x))wz=(h2(x)y+h1(x)z+h0(x))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]];
 

{{w(x,y,z)c1(yexp(1xf1(K[1])dK[1])1xexp(1K[2]f1(K[1])dK[1])f2(K[2])dK[2],zexp(1xg1(K[3])dK[3])1xexp(1K[4]g1(K[3])dK[3])g2(K[4])dK[4])exp(1x(h0(K[5])+exp(1K[5]f1(K[1])dK[1])h2(K[5])(exp(1xf1(K[1])dK[1])y1xexp(1K[2]f1(K[1])dK[1])f2(K[2])dK[2]+1K[5]exp(1K[2]f1(K[1])dK[1])f2(K[2])dK[2])+exp(1K[5]g1(K[3])dK[3])h1(K[5])(exp(1xg1(K[3])dK[3])z1xexp(1K[4]g1(K[3])dK[3])g2(K[4])dK[4]+1K[5]exp(1K[4]g1(K[3])dK[3])g2(K[4])dK[4]))dK[5])}}

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(x,y,z)=_F1(ye(f1(x)dx)(e(f1(x)dx)f2(x)dx),ze(g1(x)dx)(e(g1(x)dx)g2(x)dx))ex(ye(f1(x)dx)ef1(_h)d_hh2(_h)+ze(g1(x)dx)eg1(_h)d_hh1(_h)+(e(g1(_h)d_h)g2(_h)d_h)eg1(_h)d_hh1(_h)+(e(f1(_h)d_h)f2(_h)d_h)ef1(_h)d_hh2(_h)(e(f1(x)dx)f2(x)dx)ef1(_h)d_hh2(_h)(e(g1(x)dx)g2(x)dx)eg1(_h)d_hh1(_h)+h0(_h))d_h

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7.8.23.6 [1898] Problem 6

problem number 1898

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) wx+(y2a2+aλsinh(λx)a2sinh2(λx))wy+f(x)sinh(γz)wz=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]];
 

{{w(x,y,z)exp(1xg(K[3])dK[3])c1(log(tanh(γz2))γ1xf(K[2])dK[2],2λeaeλx(e2λx1)λ+λxae2λx+a2yeλx1eλxea(K[1]21)λK[1]K[1]dK[1])}}

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(x,y,z)=_F1(2(((sinh2(λx))2+isinh(λx)+12)λHeunCPrime(4iaλ,12,12,2iaλ,8ia3λ8λ,isinh(λx)2+12)cosh(λx)+(i(sinh2(λx))+2sinh(λx)i)(acosh(λx)+y)HeunC(4iaλ,12,12,2iaλ,8ia3λ8λ,isinh(λx)2+12))sinh(λx)+i(sinh(λx)+i)(sinh2(λx)+1)λHeunCPrime(4iaλ,12,12,2iaλ,8ia3λ8λ,isinh(λx)2+12)cosh(λx)+(2(isinh(λx)+1)(sinh2(λx)+1)y+(2ia(sinh3(λx))+(iλ+2a)(sinh2(λx))+2aiλ+(2ia+2λ)sinh(λx))cosh(λx))HeunC(4iaλ,12,12,2iaλ,8ia3λ8λ,isinh(λx)2+12),γ(f(x)dx)2arctanh(eγz)γ)eg(x)dx

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7.8.23.7 [1899] Problem 7

problem number 1899

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) wx+(f1(x)y+f2(x)yk)wy+(g1(x)z+g2(x)zm)wz=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]];
 

{{w(x,y,z)exp(1xh(K[5])dK[5])c1((k1)1xexp((k1)1K[2]f1(K[1])dK[1])f2(K[2])dK[2]+y1kexp((k1)1xf1(K[1])dK[1]),(m1)1xexp((m1)1K[4]g1(K[3])dK[3])g2(K[4])dK[4]+z1mexp((m1)1xg1(K[3])dK[3]))}}

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(x,y,z)=_F1(yk+1e(k1)(f1(x)dx)+(k1)(e(k1)(f1(x)dx)f2(x)dx),zm+1e(m1)(g1(x)dx)+(m1)(e(m1)(g1(x)dx)g2(x)dx))eh(x)dx

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7.8.23.8 [1900] Problem 8

problem number 1900

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) wx+(f1(x)y+f2(x)yk)wy+(g1(x)+g2(x)eλz)wz=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(x,y,z)=_F1(yk+1e(k1)(f1(x)dx)+(k1)(e(k1)(f1(x)dx)f2(x)dx),λ(eλ(g1(x)dx)g2(x)dx)e(z(g1(x)dx))λλ)eh(x)dx

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7.8.23.9 [1901] Problem 9

problem number 1901

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) wx+(f1(x)+f2(x)eλy)wy+(g1(x)+g2(x)eβz)wz=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(x,y,z)=_F1(λ(eλ(f1(x)dx)f2(x)dx)e(y(f1(x)dx))λλ,β(eβ(g1(x)dx)g2(x)dx)e(z(g1(x)dx))ββ)eh(x)dx

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