104 HFOPDE, chapter 4.3.2

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104.1 Problem 1

problem number 878

Added Feb. 23, 2019.

Problem Chapter 4.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = (c y e^{\lambda x} + k x e^{\mu y} ) w \]

Mathematica ✓

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

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a y-b x}{a}\right ) \exp \left (\frac {k e^{\frac {b \mu x}{a}} \left (\frac {a x e^{\frac {\mu (a y-b x)}{a}}}{b \mu }-\frac {a^2 e^{\frac {\mu (a y-b x)}{a}}}{b^2 \mu ^2}\right )}{a}+\frac {b c e^{\lambda x} \left (\frac {x}{\lambda }-\frac {1}{\lambda ^2}\right )}{a^2}+\frac {c e^{\lambda x} (a y-b x)}{a^2 \lambda }\right )\right \}\right \} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; 
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*y*exp(lambda*x) + k*x*exp(mu*y))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{{\frac {1}{{a}^{2}{\lambda }^{2}{\mu }^{2}{b}^{2}} \left ( \left ( ya-bx \right ) c{{\rm e}^{\lambda \,x}}\lambda \,{\mu }^{2}{b}^{2}+{{\rm e}^{\lambda \,x}}{b}^{3}c\lambda \,{\mu }^{2}x+kx{{\rm e}^{{\frac { \left ( ya-bx \right ) \mu }{a}}+{\frac {b\mu \,x}{a}}}}b\mu \,{a}^{2}{\lambda }^{2}-{{\rm e}^{\lambda \,x}}c{b}^{3}{\mu }^{2}-k{a}^{3}{{\rm e}^{{\frac { \left ( ya-bx \right ) \mu }{a}}+{\frac {b\mu \,x}{a}}}}{\lambda }^{2} \right ) }}} \]

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104.2 Problem 2

problem number 879

Added Feb. 23, 2019.

Problem Chapter 4.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a x e^{\lambda x+\mu y} w \]

Mathematica ✓

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

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

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

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

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104.3 Problem 3

problem number 880

Added Feb. 23, 2019.

Problem Chapter 4.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = (a y e^{\lambda x}+ b x e^{\mu y}) w \]

Mathematica ✓

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

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

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

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y}{x}} \right ) {{\rm e}^{{\frac {x}{\lambda \,\mu \,y} \left ( {\frac {a{{\rm e}^{\lambda \,x}}{y}^{2}\mu }{{x}^{2}}}+{{\rm e}^{\mu \,y}}b\lambda \right ) }}} \]

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104.4 Problem 4

problem number 881

Added Feb. 23, 2019.

Problem Chapter 4.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^k w_x + b e^{\lambda y} w_y = (c x^n+s) w \]

Mathematica ✓

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

\[ \left \{\left \{w(x,y)\to e^{\frac {x^{1-k} \left (\frac {c x^n}{-k+n+1}+\frac {s}{1-k}\right )}{a}} c_1\left (\frac {x^{-k} e^{-\lambda y} \left (a x^k-a k x^k+b \lambda x e^{\lambda y}\right )}{a (k-1) \lambda }\right )\right \}\right \} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; 
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^k*diff(w(x,y),x)+b*exp(lambda*y)*diff(w(x,y),y) =    (c*x^n+s)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{x}^{1-k}\lambda \,b-ak{{\rm e}^{-y\lambda }}+a{{\rm e}^{-y\lambda }}}{\lambda \,b \left ( k-1 \right ) }} \right ) {{\rm e}^{-{\frac {{x}^{1-k} \left ( {x}^{n}ck-{x}^{n}c+ks-sn-s \right ) }{a \left ( k-1 \right ) \left ( -n-1+k \right ) }}}} \]

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104.5 Problem 5

problem number 882

Added Feb. 23, 2019.

Problem Chapter 4.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^k w_x + b e^{\lambda x} w_y = (c e^{\mu x}+s) w \]

Mathematica ✓

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

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}{a (k+1) \lambda }\right ) \exp \left (\frac {c e^{\mu x} \left (\frac {b (k+1) e^{\lambda x}}{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}+1\right )^{\frac {k}{k+1}} \left (\left (\frac {(k+1) \left (\frac {a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}{k+1}+b e^{\lambda x}\right )}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k} \text {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {\mu }{\lambda },\frac {\lambda +\mu }{\lambda },-\frac {b (k+1) e^{\lambda x}}{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}\right )}{a \mu }-\frac {(k+1) s \left (\left (\frac {(k+1) \left (\frac {a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}{k+1}+b e^{\lambda x}\right )}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k} \left (\frac {e^{-\lambda x} \left (a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}\right )}{b (k+1)}+1\right )^{\frac {k}{k+1}} \text {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {k}{k+1},\frac {k}{k+1}+1,-\frac {e^{-\lambda x} \left (a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}\right )}{b (k+1)}\right )}{a k \lambda }\right )\right \}\right \} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; 
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*y^k*diff(w(x,y),x)+b*exp(lambda*x)*diff(w(x,y),y) =   (c*exp(mu*x)+s)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {{{\rm e}^{\lambda \,x}}bk-{y}^{k}ya\lambda +{{\rm e}^{\lambda \,x}}b}{a\lambda }} \right ) {{\rm e}^{\int ^{x}\!{\frac {c{{\rm e}^{\mu \,{\it \_a}}}+s}{a} \left ( \left ( {\frac {{{\rm e}^{\lambda \,{\it \_a}}}bk-{{\rm e}^{\lambda \,x}}bk+{y}^{k}ya\lambda -{{\rm e}^{\lambda \,x}}b+{{\rm e}^{\lambda \,{\it \_a}}}b}{a\lambda }} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}}{d{\it \_a}}}} \]

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104.6 Problem 6

problem number 883

Added Feb. 23, 2019.

Problem Chapter 4.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda x} w_x + b y^k w_y = (c x^n+s) w \]

Mathematica ✓

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

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {y^{-k} e^{-\lambda x} \left (a \lambda y e^{\lambda x}+b y^k-b k y^k\right )}{a (k-1) \lambda }\right ) \exp \left (-\frac {c x^n (\lambda x)^{-n} \text {Gamma}(n+1,\lambda x)}{a \lambda }-\frac {s e^{-\lambda x}}{a \lambda }\right )\right \}\right \} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; 
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*exp(lambda*x)*diff(w(x,y),x)+b*y^k*diff(w(x,y),y) =   (c*x^n+s)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{y}^{1-k}a\lambda -{{\rm e}^{-\lambda \,x}}bk+b{{\rm e}^{-\lambda \,x}}}{a\lambda }} \right ) {{\rm e}^{{\frac {c{x}^{n} \left ( \lambda \,x \right ) ^{-n/2}{{\rm e}^{-1/2\,\lambda \,x}} \WhittakerM \left ( n/2,n/2+1/2,\lambda \,x \right ) -{{\rm e}^{-\lambda \,x}}ns-{{\rm e}^{-\lambda \,x}}s+sn+s}{ \left ( n+1 \right ) a\lambda }}}} \]

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104.7 Problem 7

problem number 884

Added Feb. 23, 2019.

Problem Chapter 4.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda y} w_x + b x^k w_y = (c e^{\mu x}+s) w \]

Mathematica ✗

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

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

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

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {{x}^{k+1}\lambda \,b-{{\rm e}^{y\lambda }}ak-a{{\rm e}^{y\lambda }}}{ \left ( k+1 \right ) \lambda \,b}} \right ) {{\rm e}^{\int ^{x}\!{\frac { \left ( c{{\rm e}^{\mu \,{\it \_a}}}+s \right ) \left ( k+1 \right ) }{\lambda \,b} \left ( -{\frac { \left ( {x}^{k+1}\lambda \,b-{{\rm e}^{y\lambda }}ak-a{{\rm e}^{y\lambda }} \right ) k}{ \left ( k+1 \right ) \lambda \,b}}+{{\it \_a}}^{k+1}-{\frac {{x}^{k+1}\lambda \,b-{{\rm e}^{y\lambda }}ak-a{{\rm e}^{y\lambda }}}{ \left ( k+1 \right ) \lambda \,b}} \right ) ^{-1}}{d{\it \_a}}}} \]