6.8.1 2.1

6.8.1.1 [1750] Problem 1
6.8.1.2 [1751] Problem 2
6.8.1.3 [1752] Problem 3
6.8.1.4 [1753] Problem 4
6.8.1.5 [1754] Problem 5
6.8.1.6 [1755] Problem 6
6.8.1.7 [1756] Problem 7
6.8.1.8 [1757] Problem 8
6.8.1.9 [1758] Problem 9

6.8.1.1 [1750] Problem 1

problem number 1750

Added June 27, 2019.

Problem Chapter 8.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y,z)

awx+bwy+cwz=(αx+βy+γz+δ)w

Mathematica

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*D[w[x,y,z],z]== (alpha*x+beta*y+gamma*z+delta)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

{{w(x,y,z)c1(ybxa,zcxa)exp(x(a(αx+2βy+2δ+2γz)x(bβ+cγ))2a2)}}

Maple

restart; 
local gamma; 
pde :=  a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*diff(w(x,y,z),z)=  (alpha*x+beta*y+gamma*z+delta)*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(ayxba,zacxa)exa2((βy+γz+αx2+δ)ax(βb+cγ)2)

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6.8.1.2 [1751] Problem 2

problem number 1751

Added June 27, 2019.

Problem Chapter 8.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y,z)

wx+azwy+bywz=(cx+s)w

Mathematica

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

{{w(x,y,z)ecx22+sxc1(eabx(by(e2abx+1)az(e2abx1))2b,eabx(az(e2abx+1)by(e2abx1))2a)}}

Maple

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ a*z*diff(w(x,y,z),y)+ b*y*diff(w(x,y,z),z)=  (c*x+s)*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(z2aby2a,(xba+ln((aby+a2z2ba)1ba))1ba)ey((cxs)ba+c(ln((aby+a2z2ba)1ba)ln((_aab+(z2a+(_a2y2)b)aba)1ba)))1(z2a+(_a2y2)b)a1bad_a

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6.8.1.3 [1752] Problem 3

problem number 1752

Added June 27, 2019.

Problem Chapter 8.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y,z)

wx+(a1x+a0)wy+(b1x+b0)wz=(αx+βy+γz+δ)w

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a1*x+a0)*D[w[x, y,z], y] + (b1*x+b0)*D[w[x,y,z],z]== (alpha*x+beta*y+gamma*z+delta)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

{{w(x,y,z)c1(a0xa1x22+y,b0xb1x22+z)exp(16x(3a0βx2a1βx2+3αx3b0γx2b1γx2+6βy+6δ+6γz))}}

Maple

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a1*x+a0)*diff(w(x,y,z),y)+ (b1*x+b0)*diff(w(x,y,z),z)=  (alpha*x+beta*y+gamma*z+delta)*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(a1x22a0x+y,b1x22b0x+z)ex3((b1x2+3b0x23z)γ+(a1x2+3a0x23y)β3αx23δ)

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6.8.1.4 [1753] Problem 4

problem number 1753

Added June 27, 2019.

Problem Chapter 8.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y,z)

wx+(a2y+a1x+a0)wy+(b2y+b1x+b0)wz=(c2y+c1z+c0x+s)w

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a2*y+a1*x+a0)*D[w[x, y,z], y] + (b2*y+b1*x+b0)*D[w[x,y,z],z]== (c2*y+c1*z+c0*x+s)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

{{w(x,y,z)c1(ea2x(a2(a0+a2y)+a1a2x+a1)a22,ea2x(a2(2a0b2(a2xea2x+1)a2(a2ea2x(2b0x+b1x22z)+2b2y(ea2x1)))+a1b2(a22x2ea2x+2a2x+2))2a23)exp(a2(3a0b2c1(a22x22a2x+2)6a0a2c2(a2x1)+a23x(3b0c1x2b1c1x2+3c0x+6c1z+6s)+6a22y(c2b2c1x)+6a2b2c1y)+a1(b2c1(2a23x33a22x2+6)3a2c2(a22x22))6a24)}}

Maple

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+ (a2*y+a1*x+a0)*diff(w(x,y,z),y)+ (b2*y+b1*x+b0)*diff(w(x,y,z),z)=  (c2*y+c1*z+c0*x+s)*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(ea2x(ya22+a2(a1x+a0)+a1)a22,(b1x22b0x+2z)a23+b2(a1x2+2a0x2y)a222a0a2b22a1b22a23)e16a24(((2x3a1a23+(3a0a233a1a22)x2+(6a23y6a0a22)x+6ya22+6a0a2+6a1)b22xa24(b1x2+3/2b0x3z))c1+6(1/2a22(c2a1+a2c0)x2+(a0a22c2+a23s)x+c2(ya22+a0a2+a1))a2)

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6.8.1.5 [1754] Problem 5

problem number 1754

Added June 27, 2019.

Problem Chapter 8.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y,z)

wx+(ay+k1x+k0)wy+(bz+s1x+s0)wz=(c1x+c0)w

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a*y+k1*x+k0)*D[w[x, y,z], y] + (b*z+s1*x+s0)*D[w[x,y,z],z]== (c1*x+c0)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

{{w(x,y,z)ec0x+c1x22c1(eax(a2y+a(k0+k1x)+k1)a2,ebx(b2z+b(s0+s1x)+s1)b2)}}

Maple

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+(a*y+k1*x+k0)*diff(w(x,y,z),y)+(b*z+s1*x+s0)*diff(w(x,y,z),z)=  (c1*x+c0)*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(eax(ya2+a(k1x+k0)+k1)a2,exb(zb2+b(s1x+s0)+s1)b2)ex(c1x+2c0)2

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6.8.1.6 [1755] Problem 6

problem number 1755

Added June 27, 2019.

Problem Chapter 8.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y,z)

axwx+bywy+czwz=(αx+βy+γz+δ)w

Mathematica

ClearAll["Global`*"]; 
pde =  a*x*D[w[x, y,z], x] + b*y*D[w[x, y,z], y] + c*z*D[w[x,y,z],z]== (alpha*x+beta*y+gamma*z+beta)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

{{w(x,y,z)xβac1(yxba,zxca)eαxa+βyb+γzc}}

Maple

restart; 
local gamma; 
pde :=  a*x*diff(w(x,y,z),x)+ b*y*diff(w(x,y,z),y)+c*z*diff(w(x,y,z),z)=  (alpha*x+beta*y+gamma*z+beta)*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)=xβa_F1(yxba,zxca)e(yaβ+αxb)c+zγababc

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6.8.1.7 [1756] Problem 7

problem number 1756

Added June 27, 2019.

Problem Chapter 8.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y,z)

xwx+azwy+bywz=cw

Mathematica

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

{{w(x,y,z)xcc1(iysinh(ablog(x))iazcosh(ablog(x))b,ycosh(ablog(x))azsinh(ablog(x))b)}}

Maple

restart; 
local gamma; 
pde :=  x*diff(w(x,y,z),x)+ a*z*y*diff(w(x,y,z),y)+b*y*diff(w(x,y,z),z)=  c*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(z2a2bya,xe21aarctanh(z1z2a2bya)1z2a2bya)e2caarctanh(a2z2a1z2a2bya)1z2a2bya

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6.8.1.8 [1757] Problem 8

problem number 1757

Added June 27, 2019.

Problem Chapter 8.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y,z)

abxwx+b(ay+bz)wy+a(aybz)wz=cw

Mathematica

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

Failed

Maple

restart; 
local gamma; 
pde :=   a*b*x*diff(w(x,y,z),x)+ b*(a*y+b*z)*diff(w(x,y,z),y)+a*(a*y-b*z)*diff(w(x,y,z),z)=  c*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(1a2y2+2abyz+b2z2,x((ya22a2y2+2abyz+b2z2+(ay1a2y2+2abyz+b2z2+bz1a2y2+2abyz+b2z2)a2a2y2+2abyz+b2z2)1a2a2y2+2abyz+b2z2)a221a2y2+2abyz+b2z21a2a2y2+2abyz+b2z2)(ya22a2y2+2abyz+b2z21a2a2y2+2abyz+b2z2+ay1a2y2+2abyz+b2z2+bz1a2y2+2abyz+b2z2)c22b1a2y2+2abyz+b2z21a2a2y2+2abyz+b2z2

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6.8.1.9 [1758] Problem 9

problem number 1758

Added June 27, 2019.

Problem Chapter 8.2.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y,z)

(a1x+a0)wx+(b1y+b0)wy+(c1z+c0)wz=(αx+βy+γz+δ)w

Mathematica

ClearAll["Global`*"]; 
pde =  (a1*x+a0)*D[w[x, y,z], x] + (b1*y+b0)*D[w[x, y,z], y] +(c1*z+c0)*D[w[x,y,z],z]== (alpha*x+beta*y+gamma*z+delta)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

{{w(x,y,z)(a0+a1x)a0αb1c1+a1b0βc1+a1b1c0γa1b1c1δa12b1c1c1((b0+b1y)(a0+a1x)b1a1b1,(c0+c1z)(a0+a1x)c1a1c1)exp(αxa1+β(b0+b1y)b12+γ(c0+c1z)c12)}}

Maple

restart; 
local gamma; 
pde :=   (a1*x+a0)*diff(w(x,y,z),x)+(b1*y+b0)*diff(w(x,y,z),y)+(c1*z+c0)*diff(w(x,y,z),z)=  (alpha*x+beta*y+gamma*z+delta)*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(b1y+b0b1(a1x+a0)b1a1,c1z+c0c1(a1x+a0)c1a1)(a1x+a0)a0αa12b0βa1b1c0γa1c1+δa1eαxa1+β(b1y+b0)b12+(c1z+c0)γc12

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