43 HFOPDE, chapter 2.3.1

____________________________________________________________________________________

43.1 problem number 1

problem number 342

Added January 2, 2019.

Problem 2.3.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + a e^{\lambda x} w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; 
 pde = D[w[x, y], x] + a*Exp[lambda*x]*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\lambda y-a e^{\lambda x}}{\lambda }\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; 
pde := diff(w(x,y),x)+ a*exp(lambda*x)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {a{{\rm e}^{\lambda \,x}}-y\lambda }{\lambda }} \right ) \]

____________________________________________________________________________________

43.2 problem number 2

problem number 343

Added January 7, 2019.

Problem 2.3.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a e^{\lambda x} +b \right ) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; 
 pde = D[w[x, y], x] + (a*Exp[lambda*x] + b)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a \left (-e^{\lambda x}\right )-b \lambda x+\lambda y}{\lambda }\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; 
pde := diff(w(x,y),x)+ (a*exp(lambda*x)+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {bx\lambda +a{{\rm e}^{\lambda \,x}}-y\lambda }{\lambda }} \right ) \]

____________________________________________________________________________________

43.3 problem number 3

problem number 344

Added January 7, 2019.

Problem 2.3.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a e^{\lambda y} +b \right ) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; 
 pde = D[w[x, y], x] + (a*Exp[lambda*y] + b)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\log \left (\frac {e^{\lambda y}}{a e^{\lambda y}+b}\right )-b \lambda x}{b \lambda }\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; 
pde := diff(w(x,y),x)+ (a*exp(lambda*y)+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {\ln \left ( b{{\rm e}^{bx\lambda -y\lambda }}+{{\rm e}^{bx\lambda }}a \right ) }{\lambda \,b}} \right ) \]

____________________________________________________________________________________

43.4 problem number 4

problem number 345

Added January 7, 2019.

Problem 2.3.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a e^{\lambda y+ \beta x} +b \right ) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; 
 pde = D[w[x, y], x] + (a*Exp[lambda*y + beta*x] + b)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\log \left (a \lambda e^{b \lambda x+\beta x}+\beta e^{\lambda (b x-y)}+b \lambda e^{\lambda (b x-y)}\right )}{b \lambda +\beta }\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; 
pde := diff(w(x,y),x)+ (a*exp(lambda*y+beta*x)+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {-bx\lambda +y\lambda +\ln \left ( \left ( a\lambda \,{{\rm e}^{\beta \,x+y\lambda }}+\lambda \,b+\beta \right ) ^{-1} \right ) }{\lambda \,b+\beta }} \right ) \]

____________________________________________________________________________________

43.5 problem number 5

problem number 346

Added January 7, 2019.

Problem 2.3.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a e^{\lambda y+ \beta x} +b e^{\gamma x}\right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; 
 pde = D[w[x, y], x] + (a*Exp[lambda*y + beta*x] + b*Exp[gamma*x])*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; 
pde := diff(w(x,y),x)+ (a*exp(lambda*y+beta*x)+b*exp(g*x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{\lambda } \left ( {{\rm e}^{{\frac {\lambda \, \left ( b{{\rm e}^{gx}}-gy \right ) }{g}}}}+a\int \!{{\rm e}^{\beta \,x+{\frac {\lambda \,b{{\rm e}^{gx}}}{g}}}}\,{\rm d}x\lambda \right ) } \right ) \]

____________________________________________________________________________________

43.6 problem number 6

problem number 347

Added January 7, 2019.

Problem 2.3.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a e^{\lambda x} w_x + b e^{\beta y} w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; 
 pde = a*Exp[lambda*x]*D[w[x, y], x] + b*Exp[beta*y]*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\beta y-\lambda x} \left (b \beta e^{\beta y}-a \lambda e^{\lambda x}\right )}{a \beta \lambda }\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; 
pde := a*exp(lambda*x)*diff(w(x,y),x)+ b*exp(beta*y)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( {{\rm e}^{\beta \,y}}b\beta -a\lambda \,{{\rm e}^{\lambda \,x}} \right ) {{\rm e}^{-\beta \,y-\lambda \,x}}}{b\beta \,\lambda }} \right ) \]

____________________________________________________________________________________

43.7 problem number 7

problem number 348

Added January 7, 2019.

Problem 2.3.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \left ( a e^{\lambda x} +b \right ) w_x + \left ( c e^{\beta x}+d \right ) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; 
 pde = (a*Exp[lambda*x] + b)*D[w[x, y], x] + (c + Exp[beta*x] + d)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {-\lambda e^{\beta x} \text {Hypergeometric2F1}\left (1,\frac {\beta }{\lambda },\frac {\beta }{\lambda }+1,-\frac {a e^{\lambda x}}{b}\right )+\beta c \log \left (a e^{\lambda x}+b\right )+\beta d \log \left (a e^{\lambda x}+b\right )+b \beta \lambda y-\beta c \lambda x-\beta d \lambda x}{b \beta \lambda }\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; 
pde := (a*exp(lambda*x)+b)*diff(w(x,y),x)+ (c+exp(beta*x)+d)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac {c+{{\rm e}^{\beta \,x}}+d}{a{{\rm e}^{\lambda \,x}}+b}}\,{\rm d}x+y \right ) \]

____________________________________________________________________________________

43.8 problem number 8

problem number 349

Added January 7, 2019.

Problem 2.3.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \left ( a e^{\lambda x} +b \right ) w_x + \left ( c e^{\beta y}+d \right ) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; 
 pde = (a*Exp[lambda*x] + b)*D[w[x, y], x] + (c + Exp[beta*y] + d)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {\log \left (\left (c e^{\frac {\beta c x}{b}+\frac {\beta d x}{b}-\beta y}+d e^{\frac {\beta c x}{b}+\frac {\beta d x}{b}-\beta y}+e^{\frac {\beta x (c+d)}{b}}\right ) \left (a e^{\lambda x}+b\right )^{-\frac {\beta c}{b \lambda }-\frac {\beta d}{b \lambda }}\right )}{\beta (c+d)}\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; 
pde := (a*exp(lambda*x)+b)*diff(w(x,y),x)+ (c+exp(beta*y)+d)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\lambda \,b\beta \, \left ( c+d \right ) } \left ( yb\beta \,\lambda -\lambda \,x\beta \,c-\lambda \,x\beta \,d+\ln \left ( a{{\rm e}^{\lambda \,x}}+b \right ) \beta \,c+\ln \left ( a{{\rm e}^{\lambda \,x}}+b \right ) \beta \,d-\RootOf \left ( {{\rm e}^{{\frac {c{\it \_Z}}{c+d}}}} \left ( -{{\rm e}^{{\frac {y\beta \,c}{c+d}}}} \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac {\beta \,{c}^{2}}{\lambda \,b \left ( c+d \right ) }}} \left ( \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac {\beta \,cd}{\lambda \,b \left ( c+d \right ) }}} \right ) ^{2}{{\rm e}^{{\frac {y\beta \,d}{c+d}}}} \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac {\beta \,{d}^{2}}{\lambda \,b \left ( c+d \right ) }}}+ \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac {\beta \, \left ( c+d \right ) }{\lambda \,b}}}{{\rm e}^{{\frac {c{\it \_Z}}{c+d}}}}{{\rm e}^{{\frac {{\it \_Z}\,d}{c+d}}}}- \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac {\beta \, \left ( c+d \right ) }{\lambda \,b}}}c- \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac {\beta \, \left ( c+d \right ) }{\lambda \,b}}}d \right ) \right ) \lambda \,b \right ) } \right ) \] Has RootOf

____________________________________________________________________________________

43.9 problem number 9

problem number 350

Added January 7, 2019.

Problem 2.3.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \left ( a e^{\lambda y} +b \right ) w_x + \left ( c e^{\beta x}+d \right ) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; 
 pde = (a*Exp[lambda*y] + b)*D[w[x, y], x] + (c + Exp[beta*x] + d)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a \beta e^{\lambda y}+b \beta \lambda y-\beta c \lambda x-\beta d \lambda x-\lambda e^{\beta x}}{\beta \lambda }\right )\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; 
pde := (a*exp(lambda*y)+b)*diff(w(x,y),x)+ (c+exp(beta*x)+d)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {-yb\beta \,\lambda +\lambda \,x\beta \,c+\lambda \,x\beta \,d-a{{\rm e}^{y\lambda }}\beta +{{\rm e}^{\beta \,x}}\lambda }{\beta \,\lambda }} \right ) \]

____________________________________________________________________________________

43.10 problem number 10

problem number 351

Added January 7, 2019.

Problem 2.3.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \left ( a e^{\lambda x} +b e^{\beta y}\right ) w_x + a \lambda e^{\lambda x} w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; 
 pde = (a*Exp[lambda*x] + b*Exp[beta*y])*D[w[x, y], x] + a*lambda*Exp[lambda*x]*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; 
pde := (a*exp(lambda*x)+b*exp(beta*y))*diff(w(x,y),x)+ a*lambda*exp(lambda*x)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {\lambda \,x+\ln \left ( a\beta -b{{\rm e}^{\beta \,y-\lambda \,x}}-a \right ) -y}{\beta -1}} \right ) \]

____________________________________________________________________________________

43.11 problem number 11

problem number 352

Added January 7, 2019.

Problem 2.3.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \left ( a e^{\lambda x+\beta y} +c \mu \right ) w_x - \left ( b e^{\gamma x+ mu y}+c \lambda \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu]; 
 pde = (a*Exp[lambda*x + beta*y] + c*mu)*D[w[x, y], x] - (b*Exp[gamma*x + mu*y] + c*lambda)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

Maple ✗

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu'; 
pde :=  (a*exp(lambda*x+beta*y)+c*mu)*diff(w(x,y),x)- (b*exp(g*x+ mu*y)+c*lambda)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { sol=() } \]