6.4.13 5.1

6.4.13.1 [1102] Problem 1

problem number 1102

Added Feb. 25, 2019.

Problem Chapter 4.5.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {y a -x b}{a}\right ) \left (\beta y +\lambda x \right )^{\frac {c \left (\beta y +\lambda x \right )}{\lambda a +b \beta }} {\mathrm e}^{-\frac {c \left (\beta y +\lambda x \right )}{\lambda a +b \beta }}\]

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6.4.13.2 [1103] Problem 2

problem number 1103

Added Feb. 25, 2019.

Problem Chapter 4.5.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \left ( c \ln (\lambda x)+ k \ln (\beta y) \right ) w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {a y -b x}{a}\right ) \left (\lambda x \right )^{\frac {c x}{a}} \left (\beta y \right )^{\frac {k y}{b}} {\mathrm e}^{\frac {-k a y -b c x}{a b}}\]

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6.4.13.3 [1104] Problem 3

problem number 1104

Added Feb. 25, 2019.

Problem Chapter 4.5.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \ln ^n(\lambda x) w_y = \left ( c \ln ^m(\mu x)+ s \ln ^k(\beta y) \right ) w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.13.4 [1105] Problem 4

problem number 1105

Added Feb. 25, 2019.

Problem Chapter 4.5.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \ln ^n(\lambda y) w_y = \left ( c \ln ^m(\mu x)+ s \ln ^k(\beta y) \right ) w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.13.5 [1106] Problem 5

problem number 1106

Added Feb. 25, 2019.

Problem Chapter 4.5.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \ln (\beta y) w_x + a \ln (\lambda x) w_y = b w \ln (\beta y) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to e^{b x} c_1\left (y \left (\log \left (\beta y e^{\frac {a x}{y}} x^{-\frac {a x}{y}} \lambda ^{-\frac {a x}{y}}\right )-1\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {-x \ln \left (\lambda x \right ) a +y \ln \left (\beta y \right )+x a -y}{a}\right ) {\mathrm e}^{b x}\]

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6.4.13.6 [1107] Problem 6

problem number 1107

Added Feb. 25, 2019.

Problem Chapter 4.5.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \ln (\lambda x)^n w_x + b \ln (\beta y)^k w_y = c \ln (\gamma x)^m w \]

Mathematica ✗

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

Failed

Maple ✓

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

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

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