125 HFOPDE, chapter 5.2.1

____________________________________________________________________________________

125.1 Problem 1

problem number 1005

Added March 10, 2019.

Problem Chapter 5.2.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c w + d \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x, y] + d; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';t:='t'; 
v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  a*diff(w(x,y),x)+b*diff(w(x,y),y) = c*w(x,y)+d; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\frac {1}{c} \left ( {{\rm e}^{{\frac {cx}{a}}}}{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) c-d \right ) } \]

____________________________________________________________________________________

125.2 Problem 2

problem number 1006

Added March 10, 2019.

Problem Chapter 5.2.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (x-a) w_x + (y-b) w_y = w - c \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = (x - a)*D[w[x, y], x] + (y - b)*D[w[x, y], y] == w[x, y] - c; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';t:='t'; 
v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  (x-a)*diff(w(x,y),x)+(y-b)*diff(w(x,y),y) = w(x,y)-c; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-b+y}{a-x}} \right ) a-{\it \_F1} \left ( {\frac {-b+y}{a-x}} \right ) x+c \]

____________________________________________________________________________________

125.3 Problem 3

problem number 1007

Added March 10, 2019.

Problem Chapter 5.2.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x+b) w_x + (c x+d) w_y = \alpha w + \beta \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = (a*x + b)*D[w[x, y], x] + (c*x + d)*D[w[x, y], y] == alpha*w[x, y] + beta; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \frac {\alpha (a x+b)^{\frac {\alpha }{a}} c_1\left (\frac {a^2 y+b c \log (a x+b)-a d \log (a x+b)-a c x}{a^2}\right )-\beta }{\alpha }\right \}\right \} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';t:='t'; 
v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  (a*x+b)*diff(w(x,y),x)+ (c*x+d)*diff(w(x,y),y) =  alpha*w(x,y)+beta; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\frac {1}{\alpha } \left ( \left ( ax+b \right ) ^{{\frac {\alpha }{a}}}{\it \_F1} \left ( -{\frac {\ln \left ( ax+b \right ) da-\ln \left ( ax+b \right ) bc-y{a}^{2}+cxa}{{a}^{2}}} \right ) \alpha -\beta \right ) } \]

____________________________________________________________________________________

125.4 Problem 4

problem number 1008

Added March 10, 2019.

Problem Chapter 5.2.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x+b) w_x + (c y+d) w_y = \alpha w + \beta \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = (a*x + b)*D[w[x, y], x] + (c*y + d)*D[w[x, y], y] == alpha*w[x, y] + beta; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';t:='t'; 
v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; 
h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  (a*x+b)*diff(w(x,y),x)+ (c*y+d)*diff(w(x,y),y) =  alpha*w(x,y)+beta; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\frac {1}{\alpha } \left ( \left ( ax+b \right ) ^{{\frac {\alpha }{a}}}{\it \_F1} \left ( {\frac {cy+d}{c} \left ( ax+b \right ) ^{-{\frac {c}{a}}}} \right ) \alpha -\beta \right ) } \]

____________________________________________________________________________________

125.5 Problem 5

problem number 1009

Added March 10, 2019.

Problem Chapter 5.2.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x+b) w_x + (c y+d) w_y = \alpha w + \beta y+\gamma x \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = (a*x + b)*D[w[x, y], x] + (c*y + d)*D[w[x, y], y] == alpha*w[x, y] + beta*y + gamma*x; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to \frac {\alpha (a-\alpha ) (\alpha -c) (a x+b)^{\frac {\alpha }{a}} c_1\left (\frac {(c y+d) (a x+b)^{-\frac {c}{a}}}{c}\right )-a \beta (\alpha y+d)+\alpha ^2 \beta y+\alpha ^2 \gamma x+\alpha b \gamma +\alpha \beta d-\alpha c \gamma x-b c \gamma }{\alpha (a-\alpha ) (\alpha -c)}\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  (a*x+b)*diff(w(x,y),x)+ (c*y+d)*diff(w(x,y),y) =  alpha*w(x,y)+beta*y+gamma*x; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol);
 

\[ w \left ( x,y \right ) ={\frac {1}{ \left ( -\alpha +c \right ) \left ( a-\alpha \right ) \alpha } \left ( \left ( ax+b \right ) ^{{\frac {\alpha }{a}}}\alpha \, \left ( -\alpha +c \right ) \left ( a-\alpha \right ) {\it \_F1} \left ( {\frac {cy+d}{c} \left ( ax+b \right ) ^{-{\frac {c}{a}}}} \right ) + \left ( -\beta \,y-\gamma \,x \right ) {\alpha }^{2}+ \left ( ya\beta +\gamma \,xc-b\gamma -\beta \,d \right ) \alpha +a\beta \,d+bc\gamma \right ) } \]

____________________________________________________________________________________

125.6 Problem 6

problem number 1010

Added March 10, 2019.

Problem Chapter 5.2.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x+b) w_x + (c x+d y) w_y = \alpha w + \beta \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = (a*x + b)*D[w[x, y], x] + (c*x + d*y)*D[w[x, y], y] == alpha*w[x, y] + beta; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \frac {\alpha (a x+b)^{\frac {\alpha }{a}} c_1\left (\frac {(a x+b)^{-\frac {d}{a}} \left (a d y-b c-c d x+d^2 (-y)\right )}{d (a-d)}\right )-\beta }{\alpha }\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  (a*x+b)*diff(w(x,y),x)+ (c*x+d*y)*diff(w(x,y),y) =  alpha*w(x,y)+beta; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\frac {1}{\alpha } \left ( \left ( ax+b \right ) ^{{\frac {\alpha }{a}}}{\it \_F1} \left ( {\frac {dya-cxd-{d}^{2}y-bc}{ \left ( a-d \right ) d} \left ( ax+b \right ) ^{-{\frac {d}{a}}}} \right ) \alpha -\beta \right ) } \]

____________________________________________________________________________________

125.7 Problem 7

problem number 1011

Added March 10, 2019.

Problem Chapter 5.2.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a_1 x+a_0) w_x + (b_2 y+b_1 x+b_0) w_y = (c_2 y+c_1 x+c_0) w + k_2 y+k_1 x+k_0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2]; 
 pde = (a1*x + a0)*D[w[x, y], x] + (b2*y + b1*x + b0)*D[w[x, y], y] == (c2*y + c1*x + c0)*w[x, y] + k2*y + k1*x + k0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

Maple ✗

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  (a1*x+a0)*diff(w(x,y),x)+ (b2*y+b1*x+b0)*diff(w(x,y),y) =  (c2*y+c1*x+c0)*w(x,y)+k2*y+k1*x+k0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { Exception } \] Timed out

____________________________________________________________________________________

125.8 Problem 8

problem number 1012

Added March 10, 2019.

Problem Chapter 5.2.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y w_x + (b_1 x+b_0) w_y = (c_1 x+c_0) w + s_1 x+s0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0]; 
 pde = a*y*D[w[x, y], x] + (b1*x + b0)*D[w[x, y], y] == (c1*x + c0)*w[x, y] + s1*x + s0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  a*y*diff(w(x,y),x)+ (b1*x+b0)*diff(w(x,y),y) =  (c1*x+c0)*w(x,y)+s1*x+s0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) = \left ( {\frac {a{\it b1}\,x+a{\it b0}}{\sqrt {a{\it b1}}}}+\sqrt {a{\it b1}\,{x}^{2}+ \left ( {y}^{2}a-{\it b1}\,{x}^{2}-2\,{\it b0}\,x \right ) a+2\,a{\it b0}\,x} \right ) ^{-{\frac {{\it b0}\,{\it c1}-{\it b1}\,{\it c0}}{{\it b1}\,\sqrt {a{\it b1}}}}} \left ( \int ^{x}\!{\frac {{\it \_a}\,{\it s1}+{\it s0}}{\sqrt {a \left ( {{\it \_a}}^{2}{\it b1}+{y}^{2}a-{\it b1}\,{x}^{2}+2\,{\it \_a}\,{\it b0}-2\,{\it b0}\,x \right ) }} \left ( {\frac {a{\it b1}\,{\it \_a}+\sqrt {a \left ( {{\it \_a}}^{2}{\it b1}+{y}^{2}a-{\it b1}\,{x}^{2}+2\,{\it \_a}\,{\it b0}-2\,{\it b0}\,x \right ) }\sqrt {a{\it b1}}+a{\it b0}}{\sqrt {a{\it b1}}}} \right ) ^{{\frac {{\it b0}\,{\it c1}-{\it b1}\,{\it c0}}{{\it b1}\,\sqrt {a{\it b1}}}}}{{\rm e}^{-{\frac {{\it c1}\,\sqrt {a \left ( {{\it \_a}}^{2}{\it b1}+{y}^{2}a-{\it b1}\,{x}^{2}+2\,{\it \_a}\,{\it b0}-2\,{\it b0}\,x \right ) }}{a{\it b1}}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {{y}^{2}a-{\it b1}\,{x}^{2}-2\,{\it b0}\,x}{a}} \right ) \right ) {{\rm e}^{{\frac {\sqrt {a{\it b1}\,{x}^{2}+ \left ( {y}^{2}a-{\it b1}\,{x}^{2}-2\,{\it b0}\,x \right ) a+2\,a{\it b0}\,x}{\it c1}}{a{\it b1}}}}} \]