7.4.21 7.2

7.4.21.1 [1147] Problem 1
7.4.21.2 [1148] Problem 2
7.4.21.3 [1149] Problem 3
7.4.21.4 [1150] Problem 4
7.4.21.5 [1151] Problem 5

7.4.21.1 [1147] Problem 1

problem number 1147

Added March 9, 2019.

Problem Chapter 4.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y)

awx+bwy=(carccos(xλ+karccos(yβ))w

Mathematica

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

{{w(x,y)c1(ybxa)exp(k(a2(β2y2)(aybx)tan1(aya2(β2y2))+a2(β2y2))bβ1y2β2+akxcos1(yβ)acλ1x2λ2+acxcos1(xλ)a2)}}

Maple

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c*arccos(x/lambda)+k*arccos(y/beta))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

w(x,y)=_F1(aybxa)eakyarccos(yβ)+bcxarccos(xλ)β2y2β2aβkx2λ2+1bcλab

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7.4.21.2 [1148] Problem 2

problem number 1148

Added March 9, 2019.

Problem Chapter 4.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y)

awx+bwy=carccos(λx+βy)w

Mathematica

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

{{w(x,y)c1(ybxa)exp(c(β(bxay)sin1(βy+λx)+x(aλ+bβ)cos1(βy+λx)+a(β2y22βλxyλ2x2+1))a(aλ+bβ))}}

Maple

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*arccos(lambda*x+beta*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

w(x,y)=_F1(aybxa)e((βy+λx)arccos(βy+λx)β2y22βλxyλ2x2+1)caλ+bβ

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7.4.21.3 [1149] Problem 3

problem number 1149

Added March 9, 2019.

Problem Chapter 4.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y)

awx+bwy=axarccos(λx+βy)w

Mathematica

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

{{w(x,y)c1(ybxa)exp((a2+2β2(bxay)2)sin1(βy+λx)aβ2y22βλxyλ2x2+1(3aβy+aλx+4bβx)+2x2(aλ+bβ)2cos1(βy+λx)4(aλ+bβ)2)}}

Maple

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = a*x*arccos(lambda*x+beta*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

w(x,y)=_F1(aybxa)e(aarcsin(βy+λx)2+(2bβx+(βy+λx)a)(βy+λx)arccos(βy+λx)+(2bβx+(3βy2λx2)a)β2y22βλxyλ2x2+1)a2(aλ+bβ)2

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7.4.21.4 [1150] Problem 4

problem number 1150

Added March 9, 2019.

Problem Chapter 4.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y)

awx+barccosn(λx)wy=(carccosm(μx)+sarccosk(βy))w

Mathematica

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*ArcCos[lambda*x]^n*D[w[x, y], y] == (c*ArcCos[mu*x]^m + s*ArcCos[beta*y]^k)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

{{w(x,y)c1(y1xbcos1(λK[1])nadK[1])exp(1xscos1(β(y1xbcos1(λK[1])nadK[1]+1K[2]bcos1(λK[1])nadK[1]))k+ccos1(μK[2])madK[2])}}

Maple

restart; 
pde :=  a*diff(w(x,y),x)+ b*arccos(lambda*x)^n*diff(w(x,y),y) =(c*arccos(mu*x)^m+s*arccos(beta*y)^k)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

w(x,y)=_F1((LommelS1(n+32,32,arccos(λx))arccos(λx)+arccos(λx)n+1+(n+2)LommelS1(n+12,12,arccos(λx))arccos(λx))λ2x2+1b2n2n+(n+2)(bx2n2nLommelS1(n+12,12,arccos(λx))arccos(λx)+ay)λ(n+2)aλ)excarccos(_bμ)m+s(arccos(((LommelS1(n+32,32,arccos(_bλ))arccos(_bλ)+arccos(_bλ)n+1+(n+2)LommelS1(n+12,12,arccos(_bλ))arccos(_bλ))_b2λ2+1b2n2n+(n+2)(_bb2n2nLommelS1(n+12,12,arccos(_bλ))arccos(_bλ)ay+a(barccos(λx)nadx))λ)β(n+2)aλ)+π)kad_b

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7.4.21.5 [1151] Problem 5

problem number 1151

Added March 9, 2019.

Problem Chapter 4.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y)

awx+barccosn(λy)wy=(carccosm(μx)+sarccosk(βy))w

Mathematica

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*ArcCos[lambda*y]^n*D[w[x, y], y] == (c*ArcCos[mu*x]^m + s*ArcCos[beta*y]^k)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

{{w(x,y)c1(1ycos1(λK[1])ndK[1]bxa)exp(1ycos1(λK[2])n(scos1(βK[2])k+ccos1(μ(bxa1ycos1(λK[1])ndK[1]+a1K[2]cos1(λK[1])ndK[1])b)m)bdK[2])}}

Maple

restart; 
pde :=  a*diff(w(x,y),x)+ b*arccos(lambda*y)^n*diff(w(x,y),y) =(c*arccos(mu*x)^m+s*arccos(beta*y)^k)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

w(x,y)=_F1((LommelS1(n+32,32,arccos(λy))arccos(λy)arccos(λy)n+1+(n2)LommelS1(n+12,12,arccos(λy))arccos(λy))λ2y2+1a2n2n(n2)(ay2n2nLommelS1(n+12,12,arccos(λy))arccos(λy)bx)λ(n2)bλ)ey(carccos(((LommelS1(n+32,32,arccos(_bλ))arccos(_bλ)arccos(_bλ)n+1+(n2)LommelS1(n+12,12,arccos(_bλ))arccos(_bλ))_b2λ2+1a2n2n+(n2)(_ba2n2nLommelS1(n+12,12,arccos(_bλ))arccos(_bλ)a(arccos(λy)ndy)+bx)λ)μ(n2)bλ)m+sarccos(_bβ)k)arccos(_bλ)nbd_b

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