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'));
 

\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}{\mathrm e}^{-\frac {\textit {\_a}}{a}} \left (\operatorname {c2} \ln \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \beta }{a}\right )^{n}+\operatorname {c1} \,\textit {\_a}^{k}\right )d \textit {\_a}}{a}+f_{1} \left (\frac {a y -x b}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]

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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'));
 

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

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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'));
 

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

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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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\frac {c x \left (\left (y^{-k-1}\right )^{-\frac {1}{k+1}}\right )^{-k} \left (\frac {a (n+1) y^{k+1}}{a (n+1) y^{k+1}-b (k+1) x^{n+1}}\right )^{\frac {k}{k+1}} \operatorname {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {1}{n+1},1+\frac {1}{n+1},\frac {b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )}{a}\right ) \left (\int _1^x\frac {\exp \left (-\frac {c \operatorname {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {1}{n+1},1+\frac {1}{n+1},\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right ) K[1] \left (1-\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )^{\frac {k}{k+1}} \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k}}{a}\right ) s \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k} \log ^m(\beta K[1])}{a}dK[1]+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b x^{n+1}}{a n+a}\right )\right )\right \}\right \}\]

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'));
 

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

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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'));
 

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

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