6.5.13 5.2

6.5.13.1 [1282] Problem 1

problem number 1282

Added April 8, 2019.

Problem Chapter 5.5.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = w + c_1 x^k+ c_2 \ln ^n(\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+c1*x^k+c2*Log[beta*y]^n; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} K[1]^k+\text {c2} \log ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*x^k+c2*ln(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {{\it \_a}}{a}}}} \left ( {\it c2}\, \left ( \ln \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}+{\it c1}\,{{\it \_a}}^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {x}{a}}}}\]

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6.5.13.2 [1283] Problem 2

problem number 1283

Added April 8, 2019.

Problem Chapter 5.5.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = c w + x^k \ln ^n(\beta y) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} K[1]^k \log ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {{{\it \_a}}^{k}}{a} \left ( \ln \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]

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6.5.13.3 [1284] Problem 3

problem number 1284

Added April 8, 2019.

Problem Chapter 5.5.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x^k w_x + b x^n w_y = c w + s \ln ^m(\beta x) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to e^{\frac {c x^{1-k}}{a-a k}} \left (\int _1^x\frac {e^{\frac {c K[1]^{1-k}}{a (k-1)}} s K[1]^{-k} \log ^m(\beta K[1])}{a}dK[1]+c_1\left (y-\frac {b x^{-k+n+1}}{a (-k)+a n+a}\right )\right )\right \}\right \}\]

Maple ✓

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

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

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6.5.13.4 [1285] Problem 4

problem number 1285

Added April 8, 2019.

Problem Chapter 5.5.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x^n w_x + b y^k w_y = c w + s \ln ^m(\beta x) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to e^{\frac {c x^{1-n}}{a-a n}} \left (\int _1^x\frac {e^{\frac {c K[1]^{1-n}}{a (n-1)}} s K[1]^{-n} \log ^m(\beta K[1])}{a}dK[1]+c_1\left (\frac {b x^{1-n}}{a (n-1)}-\frac {y^{1-k}}{k-1}\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y \right ) = \left ( \int \!{\frac {s \left ( \ln \left ( \beta \,x \right ) \right ) ^{m}{x}^{-n}}{a}{{\rm e}^{{\frac {{x}^{1-n}c}{ \left ( n-1 \right ) a}}}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {-{x}^{1-n}b \left ( k-1 \right ) +{y}^{-k+1}a \left ( n-1 \right ) }{ \left ( n-1 \right ) a}} \right ) \right ) {{\rm e}^{-{\frac {{x}^{1-n}c}{ \left ( n-1 \right ) a}}}}\]

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6.5.13.5 [1286] Problem 5

problem number 1286

Added April 8, 2019.

Problem Chapter 5.5.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x^n w_x + b \ln ^n(\lambda x) w_y = c w + s x^m \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {1}{ac \left ( m-3\,n+3 \right ) \left ( m-2\,n+2 \right ) \left ( m-n+1 \right ) } \left ( -a{{\rm e}^{{\frac {{x}^{1-n}c}{ \left ( 2\,n-2 \right ) a}}}} \left ( -{\frac {{x}^{1-n}c}{ \left ( n-1 \right ) a}} \right ) ^{{\frac {m-2\,n+2}{2\,n-2}}} \left ( -{\frac {c}{ \left ( n-1 \right ) a}} \right ) ^{{\frac {m-n+1}{n-1}}} \left ( -{\frac {c}{ \left ( n-1 \right ) a}} \right ) ^{{\frac {-m+n-1}{n-1}}}{x}^{m}s \left ( n-1 \right ) \left ( m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {2\,n-2-m}{2\,n-2}},{\frac {-m+3\,n-3}{2\,n-2}},-{\frac {{x}^{1-n}c}{ \left ( n-1 \right ) a}} \right ) +s \left ( -{\frac {{x}^{1-n}c}{ \left ( n-1 \right ) a}} \right ) ^{{\frac {m-2\,n+2}{2\,n-2}}}{x}^{m} \left ( -{\frac {c}{ \left ( n-1 \right ) a}} \right ) ^{{\frac {-m+n-1}{n-1}}} \left ( n-1 \right ) ^{2}{{\rm e}^{{\frac {{x}^{1-n}c}{ \left ( 2\,n-2 \right ) a}}}} \left ( {x}^{1-n}c+a \left ( m-2\,n+2 \right ) \right ) \left ( -{\frac {c}{ \left ( n-1 \right ) a}} \right ) ^{{\frac {m-n+1}{n-1}}} \WhittakerM \left ( -{\frac {m}{2\,n-2}},{\frac {-m+3\,n-3}{2\,n-2}},-{\frac {{x}^{1-n}c}{ \left ( n-1 \right ) a}} \right ) +{\it \_F1} \left ( -{\frac {b\int \!{x}^{-n} \left ( \ln \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x}{a}}+y \right ) ac \left ( m-3\,n+3 \right ) \left ( m-n+1 \right ) \left ( m-2\,n+2 \right ) \right ) {{\rm e}^{-{\frac {{x}^{1-n}c}{ \left ( n-1 \right ) a}}}}}\]

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6.5.13.6 [1287] Problem 6

problem number 1287

Added April 8, 2019.

Problem Chapter 5.5.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a y^k w_x + b x^n w_y = c w + s \ln ^m(\beta x) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \exp \left (\frac {c x \left (\left (y^{-k-1}\right )^{-\frac {1}{k+1}}\right )^{-k} \left (\frac {a (n+1) y^{k+1}}{a (n+1) y^{k+1}-b (k+1) x^{n+1}}\right )^{\frac {k}{k+1}} \, _2F_1\left (\frac {k}{k+1},\frac {1}{n+1};1+\frac {1}{n+1};\frac {b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )}{a}\right ) \left (\int _1^x\frac {\exp \left (-\frac {c \, _2F_1\left (\frac {k}{k+1},\frac {1}{n+1};1+\frac {1}{n+1};\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right ) K[1] \left (1-\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )^{\frac {k}{k+1}} \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k}}{a}\right ) s \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k} \log ^m(\beta K[1])}{a}dK[1]+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b x^{n+1}}{a n+a}\right )\right )\right \}\right \}\]

Maple ✓

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

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

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6.5.13.7 [1288] Problem 7

problem number 1288

Added April 8, 2019.

Problem Chapter 5.5.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a y^k w_x + b \ln ^n(\lambda x) w_y = c w + s x^m \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \left (\left (y^{k+1}-(k+1) \int _1^x\frac {b \log ^n(\lambda K[1])}{a}dK[1]+(k+1) \int _1^{K[2]}\frac {b \log ^n(\lambda K[1])}{a}dK[1]\right ){}^{\frac {1}{k+1}}\right ){}^{-k}}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \left (\left (y^{k+1}-(k+1) \int _1^x\frac {b \log ^n(\lambda K[1])}{a}dK[1]+(k+1) \int _1^{K[2]}\frac {b \log ^n(\lambda K[1])}{a}dK[1]\right ){}^{\frac {1}{k+1}}\right ){}^{-k}}{a}dK[2]\right ) s K[3]^m \left (\left (y^{k+1}-(k+1) \int _1^x\frac {b \log ^n(\lambda K[1])}{a}dK[1]+(k+1) \int _1^{K[3]}\frac {b \log ^n(\lambda K[1])}{a}dK[1]\right ){}^{\frac {1}{k+1}}\right ){}^{-k}}{a}dK[3]+c_1\left (\frac {y^{k+1}}{k+1}-\int _1^x\frac {b \log ^n(\lambda K[1])}{a}dK[1]\right )\right )\right \}\right \}\]

Maple ✓

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

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

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