7.5.13 5.2

7.5.13.1 [1284] Problem 1
7.5.13.2 [1285] Problem 2
7.5.13.3 [1286] Problem 3
7.5.13.4 [1287] Problem 4
7.5.13.5 [1288] Problem 5
7.5.13.6 [1289] Problem 6
7.5.13.7 [1290] Problem 7

7.5.13.1 [1284] Problem 1

problem number 1284

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'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {c1} \,\textit {\_a}^{k}+\mathit {c2} \ln \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )^{n}\right ) {\mathrm e}^{-\frac {\textit {\_a}}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]

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7.5.13.2 [1285] Problem 2

problem number 1285

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'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\textit {\_a}^{k} \ln \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )^{n} {\mathrm e}^{-\frac {\textit {\_a} c}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]

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7.5.13.3 [1286] Problem 3

problem number 1286

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'));
 

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

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7.5.13.4 [1287] Problem 4

problem number 1287

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'));
 

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

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7.5.13.5 [1288] Problem 5

problem number 1288

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]];
 

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

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'));
 

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

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7.5.13.6 [1289] Problem 6

problem number 1289

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'));
 

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

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7.5.13.7 [1290] Problem 7

problem number 1290

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'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {s \,\textit {\_f}^{m} \left (\left (\frac {a \,y^{k +1}+\int \left (k +1\right ) b \ln \left (\textit {\_f} \lambda \right )^{n}d \textit {\_f} +\int \left (-k -1\right ) b \ln \left (\lambda x \right )^{n}d x}{a}\right )^{\frac {1}{k +1}}\right )^{-k} {\mathrm e}^{-\frac {\left (\left (k +1\right ) \textit {\_f} b +\left (a y \,y^{k}-\left (k +1\right ) b \left (\int \ln \left (\lambda x \right )^{n}d x \right )\right ) \ln \left (\textit {\_a} \lambda \right )^{-n}\right ) c \left (\left (\frac {\left (k +1\right ) \textit {\_f} b \ln \left (\textit {\_a} \lambda \right )^{n}+a \,y^{k +1}+\int \left (-k -1\right ) b \ln \left (\lambda x \right )^{n}d x}{a}\right )^{\frac {1}{k +1}}\right )^{-k}}{a b}}}{a}d \textit {\_f} +\textit {\_F1} \left (\frac {a y \,y^{k}-\left (k +1\right ) b \left (\int \ln \left (\lambda x \right )^{n}d x \right )}{a}\right )\right ) {\mathrm e}^{\frac {\left (\left (k +1\right ) b x +\left (a y \,y^{k}-\left (k +1\right ) b \left (\int \ln \left (\lambda x \right )^{n}d x \right )\right ) \ln \left (\textit {\_a} \lambda \right )^{-n}\right ) c \left (\left (\frac {\left (k +1\right ) b x \ln \left (\textit {\_a} \lambda \right )^{n}+a \,y^{k +1}+\int \left (-k -1\right ) b \ln \left (\lambda x \right )^{n}d x}{a}\right )^{\frac {1}{k +1}}\right )^{-k}}{a b}}\]

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