6.4.2 2.1

6.4.2.1 [1024] Problem 1

problem number 1024

Added Feb. 17, 2019.

Problem Chapter 4.2.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a{w}_x+b{w}_y=cw$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to e^{\frac{cx}{a}}{c}_1(y-\frac{bx}{a})\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}\left(\frac{ya-bx}{a}\right){\mathrm{e}}^{\frac{cx}{a}}$$

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6.4.2.2 [1025] Problem 2

problem number 1025

Added Feb. 17, 2019.

Problem Chapter 4.2.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a{w}_x+y{w}_y=bw$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to e^{\frac{bx}{a}}{c}_1\left(y{e}^{-\frac{x}{a}}\right)\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}\left(y{\mathrm{e}}^{-\frac{x}{a}}\right){\mathrm{e}}^{\frac{bx}{a}}$$

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6.4.2.3 [1026] Problem 3

problem number 1026

Added Feb. 17, 2019.

Problem Chapter 4.2.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$x{w}_x+y{w}_y=aw$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to x^a{c}_1\left(\frac{y}{x}\right)\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}\left(\frac{y}{x}\right)x^a$$

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6.4.2.4 [1027] Problem 4

problem number 1027

Added Feb. 17, 2019.

Problem Chapter 4.2.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$x(a{w}_x-b{w}_y)=cyw$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to e^{-bcx}{x}^{c(bx+y)}{c}_1(bx+y)\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}(bx+y)x^{(bx+y)c}{\mathrm{e}}^{-cxb}$$

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6.4.2.5 [1028] Problem 5

problem number 1028

Added Feb. 17, 2019.

Problem Chapter 4.2.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$x{w}_x+y{w}_y=axw$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to e^{ax}{c}_1\left(\frac{y}{x}\right)\}\right\}$$

Maple ✓

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

$$w(x,y)={\mathrm{e}}^{ax}\mathit{\_}\mathit{F}\mathit{1}\left(\frac{y}{x}\right)$$

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6.4.2.6 [1029] Problem 6

problem number 1029

Added Feb. 17, 2019.

Problem Chapter 4.2.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$(x-a)w_x+(y-b)w_y=w$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to -(a-x)c_1\left(\frac{b-y}{a-x}\right)\}\right\}$$

Maple ✓

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

$$w(x,y)=(a-x)\mathit{\_}\mathit{F}\mathit{1}\left(\frac{y-b}{a-x}\right)$$

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6.4.2.7 [1030] Problem 7

problem number 1030

Added Feb. 17, 2019.

Problem Chapter 4.2.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$(y+ax)w_x+(y-ax)w_y=bw$$

Mathematica ✗

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

Failed

Maple ✗

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

time expired

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