6.3.13 5.1
6.3.13.1 [924] Problem 1
problem number 924
Added Feb. 11, 2019.
Problem Chapter 3.5.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \ln (\lambda x+\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Log[lambda*x + beta*y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )+\frac {c (a \beta y-b \beta x) \log (a (\beta y+\lambda x))}{a (a \lambda +b \beta )}+\frac {c x \log (\beta y+\lambda x)}{a}-\frac {c x}{a}\right \}\right \}\]
Maple ✓
restart;
pde :=a*diff(w(x,y),x) + b*diff(w(x,y),y) = c*ln(lambda*x+beta*y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \frac {\left (\lambda a +b \beta \right ) f_{1} \left (\frac {a y -b x}{a}\right )+c \left (\ln \left (\beta y +\lambda x \right )-1\right ) \left (\beta y +\lambda x \right )}{\lambda a +b \beta }\]
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6.3.13.2 [925] Problem 2
problem number 925
Added Feb. 11, 2019.
Problem Chapter 3.5.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \ln (\lambda x) + k \ln (\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Log[lambda*x] + k*Log[beta*y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {a b c_1\left (y-\frac {b x}{a}\right )+a k y \log (\beta y)+b c x \log (\lambda x)-b c x-b k x}{a b}\right \}\right \}\]
Maple ✓
restart;
pde :=a*diff(w(x,y),x) + b*diff(w(x,y),y) = c*ln(lambda*x)+k*ln(beta*y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \frac {c x \ln \left (\lambda x \right ) b +k a y \ln \left (\beta y \right )+f_{1} \left (\frac {y a -b x}{a}\right ) a b -k a y -b c x}{a b}\]
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6.3.13.3 [926] Problem 3
problem number 926
Added Feb. 11, 2019.
Problem Chapter 3.5.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \ln (\lambda x) \ln (\beta y) w_y = c \ln (\gamma x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*Log[lambda*x]*Log[beta*y]*D[w[x, y], y] == c*Log[gamma*x];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^y\frac {c \log \left (\frac {\gamma \left (b x (\log (\lambda x)-1)-a \int _1^y\frac {1}{\log (\beta K[1])}dK[1]+a \int _1^{K[2]}\frac {1}{\log (\beta K[1])}dK[1]\right )}{b W\left (\frac {\lambda \left (b x (\log (\lambda x)-1)-a \int _1^y\frac {1}{\log (\beta K[1])}dK[1]+a \int _1^{K[2]}\frac {1}{\log (\beta K[1])}dK[1]\right )}{b e}\right )}\right )}{b \log (\beta K[2]) \log \left (\frac {\lambda \left (b x (\log (\lambda x)-1)-a \int _1^y\frac {1}{\log (\beta K[1])}dK[1]+a \int _1^{K[2]}\frac {1}{\log (\beta K[1])}dK[1]\right )}{b W\left (\frac {\lambda \left (b x (\log (\lambda x)-1)-a \int _1^y\frac {1}{\log (\beta K[1])}dK[1]+a \int _1^{K[2]}\frac {1}{\log (\beta K[1])}dK[1]\right )}{b e}\right )}\right )}dK[2]+c_1\left (\frac {a \int _1^y\frac {1}{\log (\beta K[1])}dK[1]-b x \log (\lambda x)+b x}{a}\right )\right \}\right \}\]
Maple ✓
restart;
pde :=a*diff(w(x,y),x) + b*ln(lambda*x)*ln(beta*y)*diff(w(x,y),y) = c*ln(gamma*x);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \frac {f_{1} \left (\frac {a \,\operatorname {Ei}_{1}\left (-\ln \left (\beta y \right )\right )+x b \beta \left (\ln \left (\lambda x \right )-1\right )}{\beta a}\right ) a \lambda \operatorname {LambertW}\left (\lambda x \left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1}\right )+c \left (-\operatorname {LambertW}\left (\lambda x \left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1}\right ) \left (\ln \left (\frac {\gamma x \left (\ln \left (\lambda x \right )-1\right )}{\operatorname {LambertW}\left (\lambda x \left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1}\right )}\right )-\ln \left (\frac {\lambda x \left (\ln \left (\lambda x \right )-1\right )}{\operatorname {LambertW}\left (\lambda x \left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1}\right )}\right )\right ) \left (\operatorname {LambertW}\left (\lambda x \left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1}\right )-\ln \left (\frac {\lambda x \left (\ln \left (\lambda x \right )-1\right )}{\operatorname {LambertW}\left (\lambda x \left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1}\right )}\right )+1\right ) \operatorname {Ei}_{1}\left (-\ln \left (\frac {\lambda x \left (\ln \left (\lambda x \right )-1\right )}{\operatorname {LambertW}\left (\lambda x \left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1}\right )}\right )\right )+\lambda x \left (\operatorname {LambertW}\left (\lambda x \left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1}\right )+\ln \left (\frac {\gamma x \left (\ln \left (\lambda x \right )-1\right )}{\operatorname {LambertW}\left (\lambda x \left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1}\right )}\right )-\ln \left (\frac {\lambda x \left (\ln \left (\lambda x \right )-1\right )}{\operatorname {LambertW}\left (\lambda x \left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1}\right )}\right )\right ) \left (\ln \left (\lambda x \right )-1\right )\right )}{a \lambda \operatorname {LambertW}\left (\lambda x \left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1}\right )}\]
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6.3.13.4 [927] Problem 4
problem number 927
Added Feb. 11, 2019.
Problem Chapter 3.5.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \ln ^n(\lambda x) w_y = c \ln ^m(\mu x)+ s \ln ^k(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*Log[lambda*x]^n*D[w[x, y], y] == c*Log[mu*x]^m + s*Log[beta*y]^k;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {s \log ^k\left (\beta \left (y-\int _1^x\frac {b \log ^n(\lambda K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \log ^n(\lambda K[1])}{a}dK[1]\right )\right )+c \log ^m(\mu K[2])}{a}dK[2]+c_1\left (y-\int _1^x\frac {b \log ^n(\lambda K[1])}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart;
pde :=a*diff(w(x,y),x) + b*ln(lambda*x)^n*diff(w(x,y),y) = c*ln(mu*x)^m+s*ln(beta*y)^k;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \frac {\int _{}^{x}\left (c \ln \left (\mu \textit {\_b} \right )^{m}+s {\ln \left (\frac {\beta \left (b \int \ln \left (\lambda \textit {\_b} \right )^{n}d \textit {\_b} -b \int \ln \left (\lambda x \right )^{n}d x +y a \right )}{a}\right )}^{k}\right )d \textit {\_b}}{a}+f_{1} \left (-\frac {b \int \ln \left (\lambda x \right )^{n}d x}{a}+y \right )\]
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6.3.13.5 [928] Problem 5
problem number 928
Added Feb. 11, 2019.
Problem Chapter 3.5.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \ln ^n(\lambda y) w_y = c \ln ^m(\mu x)+ s \ln ^k(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*Log[lambda*y]^n*D[w[x, y], y] == c*Log[mu*x]^m + s*Log[beta*y]^k;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^y\frac {\log ^{-n}(\lambda K[2]) \left (s \log ^k(\beta K[2])+c \log ^m\left (\frac {\mu \left (b x-a \int _1^y\log ^{-n}(\lambda K[1])dK[1]+a \int _1^{K[2]}\log ^{-n}(\lambda K[1])dK[1]\right )}{b}\right )\right )}{b}dK[2]+c_1\left (\int _1^y\log ^{-n}(\lambda K[1])dK[1]-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart;
pde :=a*diff(w(x,y),x) + b*ln(lambda*y)^n*diff(w(x,y),y) = c*ln(mu*x)^m+s*ln(beta*y)^k;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \frac {\int _{}^{y}\left (c {\ln \left (\frac {\mu \left (a \int \ln \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b} -a \int \ln \left (\lambda y \right )^{-n}d y +x b \right )}{b}\right )}^{m}+s \ln \left (\beta \textit {\_b} \right )^{k}\right ) \ln \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b}}{b}+f_{1} \left (-\frac {a \int \ln \left (\lambda y \right )^{-n}d y}{b}+x \right )\]
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6.3.13.6 [929] Problem 6
problem number 929
Added Feb. 11, 2019.
Problem Chapter 3.5.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \ln ^n(\lambda x) w_x + b \ln ^k(\beta y) w_y = c \ln ^m(\gamma x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*Log[lambda*x]^n*D[w[x, y], x] + b*Log[lambda*y]^k*D[w[x, y], y] == c*Log[gamma*x]^m;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {a c_1 \int \log ^{-k}(\lambda y) \, dy}{b}+\frac {\int \log ^{-n}(\lambda x) \left (c \log ^m(\gamma x)+a c_1\right ) \, dx}{a}+c_2\right \}\right \}\]
Maple ✓
restart;
pde :=a*ln(lambda*x)^n*diff(w(x,y),x) + b*ln(lambda*y)^k*diff(w(x,y),y) = c*ln(gamma*x)^m;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \frac {c \int \ln \left (\gamma x \right )^{m} \ln \left (\lambda x \right )^{-n}d x}{a}+f_{1} \left (-\int \ln \left (\lambda x \right )^{-n}d x +\frac {a \int \ln \left (\lambda y \right )^{-k}d y}{b}\right )\]
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