6.5.16 6.3
6.5.16.1 [1302] Problem 1
problem number 1302
Added April 11, 2019.
Problem Chapter 5.6.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + k \tan (\lambda x+\mu y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+k*Tan[lambda*x+mu*y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} c_1\left (y-\frac {b x}{a}\right )+\frac {i k \left (\operatorname {Hypergeometric2F1}\left (1,-\frac {i c}{2 a \lambda +2 b \mu },\frac {-i c+2 a \lambda +2 b \mu }{2 (a \lambda +b \mu )},-e^{-2 i (\lambda x+\mu y)}\right )-\operatorname {Hypergeometric2F1}\left (1,\frac {i c}{2 a \lambda +2 b \mu },\frac {i c+2 a \lambda +2 b \mu }{2 a \lambda +2 b \mu },-e^{2 i (\lambda x+\mu y)}\right )\right )}{c}\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) =c*w(x,y)+k*tan(lambda*x+mu*y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {k \int _{}^{x}\tan \left (\frac {\left (y a -b \left (x -\textit {\_a} \right )\right ) \mu +\textit {\_a} a \lambda }{a}\right ) {\mathrm e}^{-\frac {c \textit {\_a}}{a}}d \textit {\_a}}{a}+f_{1} \left (\frac {y a -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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6.5.16.2 [1303] Problem 2
problem number 1303
Added April 11, 2019.
Problem Chapter 5.6.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = w + c_1 \tan ^k(\lambda x) + c_2 \tan ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*Tan[lambda*x]^k + c2*Tan[beta*y]^n;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} \tan ^k(\lambda K[1])+\text {c2} \tan ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+ c1*tan(lambda*x)^k + c2*tan(beta*y)^n;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}{\mathrm e}^{-\frac {\textit {\_a}}{a}} \left (\operatorname {c1} \tan \left (\lambda \textit {\_a} \right )^{k}+\operatorname {c2} \tan \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \beta }{a}\right )^{n}\right )d \textit {\_a}}{a}+f_{1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]
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6.5.16.3 [1304] Problem 3
problem number 1304
Added April 11, 2019.
Problem Chapter 5.6.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \tan ^k(\lambda x) \tan ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ Tan[lambda*x]^k * Tan[beta*y]^n;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \tan ^k(\lambda K[1]) \tan ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ tan(lambda*x)^k *tan(beta*y)^n;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}\tan \left (\lambda \textit {\_a} \right )^{k} \tan \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \beta }{a}\right )^{n} {\mathrm e}^{-\frac {c \textit {\_a}}{a}}d \textit {\_a}}{a}+f_{1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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6.5.16.4 [1305] Problem 4
problem number 1305
Added April 11, 2019.
Problem Chapter 5.6.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \tan (\mu y) w_y = c \tan (\lambda x) w + k \tan (\nu x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*Tan[mu*y]*D[w[x, y], y] == c*Tan[lambda*x]*w[x,y]+k*Tan[nu*x];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \cos ^{-\frac {c}{a \lambda }}(\lambda x) \left (\int _1^x\frac {k \cos ^{\frac {c}{a \lambda }}(\lambda K[1]) \tan (\nu K[1])}{a}dK[1]+c_1\left (\frac {\log (\sin (\mu y))}{\mu }-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*tan(mu*y)*diff(w(x,y),y) = c*tan(lambda*x)*w(x,y)+ k*tan(nu*x);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {k \int \tan \left (\nu x \right ) \left (\sec \left (\lambda x \right )^{2}\right )^{-\frac {c}{2 a \lambda }}d x}{a}+f_{1} \left (-x +\frac {\ln \left (\operatorname {csgn}\left (\sec \left (\mu y \right )\right ) \sin \left (\mu y \right )\right ) a}{b \mu }\right )\right ) \left (\sec \left (\lambda x \right )^{2}\right )^{\frac {c}{2 a \lambda }}\]
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6.5.16.5 [1306] Problem 5
problem number 1306
Added April 11, 2019.
Problem Chapter 5.6.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = c w + k \tan (\lambda x+\nu y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*w[x,y]+k*Tan[lambda*x+nu*y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to x^{\frac {c}{a}} \left (\int _1^x\frac {k K[1]^{-\frac {a+c}{a}} \tan \left (\nu y K[1]^{\frac {b}{a}} x^{-\frac {b}{a}}+\lambda K[1]\right )}{a}dK[1]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) =c*w(x,y)+k*tan(lambda*x+nu*y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {x^{\frac {c}{a}} \left (f_{1} \left (y \,x^{-\frac {b}{a}}\right ) a +k \int _{}^{x}\tan \left (\lambda \textit {\_a} +\nu y \,x^{-\frac {b}{a}} \textit {\_a}^{\frac {b}{a}}\right ) \textit {\_a}^{\frac {-a -c}{a}}d \textit {\_a} \right )}{a}\]
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6.5.16.6 [1307] Problem 6
problem number 1307
Added April 11, 2019.
Problem Chapter 5.6.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \tan ^n(\lambda x) w_x + b \tan ^m(\mu x) w_y = c \tan ^k(\nu x) w + p \tan ^s(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*Tan[lambda*x]^n*D[w[x, y], x] + b*Tan[mu*x]^m*D[w[x, y], y] == c*Tan[nu*x]^k*w[x,y]+p*Tan[beta*y]^s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k(\nu K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k(\nu K[2])}{a}dK[2]\right ) p \tan ^{-n}(\lambda K[3]) \tan ^s\left (\beta \left (y-\int _1^x\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]+\int _1^{K[3]}\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*tan(lambda*x)^n*diff(w(x,y),x)+ b*tan(mu*x)^m*diff(w(x,y),y) =c*tan(nu*x)^k*w(x,y)+p*tan(beta*y)^s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {p \int _{}^{x}{\tan \left (\frac {\beta \left (b \int \tan \left (\mu \textit {\_f} \right )^{m} \tan \left (\lambda \textit {\_f} \right )^{-n}d \textit {\_f} -b \int \tan \left (\mu x \right )^{m} \tan \left (\lambda x \right )^{-n}d x +y a \right )}{a}\right )}^{s} \tan \left (\lambda \textit {\_f} \right )^{-n} {\mathrm e}^{-\frac {c \int \tan \left (\nu \textit {\_f} \right )^{k} \tan \left (\lambda \textit {\_f} \right )^{-n}d \textit {\_f}}{a}}d \textit {\_f}}{a}+f_{1} \left (-\frac {b \int \tan \left (\mu x \right )^{m} \tan \left (\lambda x \right )^{-n}d x}{a}+y \right )\right ) {\mathrm e}^{\frac {c \int \tan \left (\nu x \right )^{k} \tan \left (\lambda x \right )^{-n}d x}{a}}\]
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6.5.16.7 [1308] Problem 7
problem number 1308
Added April 11, 2019.
Problem Chapter 5.6.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \tan ^n(\lambda x) w_x + b \tan ^m(\mu x) w_y = c \tan ^k(\nu y) w + p \tan ^s(\beta x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*Tan[lambda*x]^n*D[w[x, y], x] + b*Tan[mu*x]^m*D[w[x, y], y] == c*Tan[nu*y]^k*w[x,y]+p*Tan[beta*x]^s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k\left (\nu \left (y-\int _1^x\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k\left (\nu \left (y-\int _1^x\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) p \tan ^s(\beta K[3]) \tan ^{-n}(\lambda K[3])}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*tan(lambda*x)^n*diff(w(x,y),x)+ b*tan(mu*x)^m*diff(w(x,y),y) =c*tan(nu*y)^k*w(x,y)+p*tan(beta*x)^s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {p \int _{}^{x}\tan \left (\beta \textit {\_b} \right )^{s} \tan \left (\lambda \textit {\_b} \right )^{-n} {\mathrm e}^{-\frac {c \int {\tan \left (\frac {\nu \left (b \int \tan \left (\lambda \textit {\_b} \right )^{-n} \tan \left (\mu \textit {\_b} \right )^{m}d \textit {\_b} -b \int \tan \left (\lambda x \right )^{-n} \tan \left (\mu x \right )^{m}d x +y a \right )}{a}\right )}^{k} \tan \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b}}{a}}d \textit {\_b}}{a}+f_{1} \left (-\frac {b \int \tan \left (\lambda x \right )^{-n} \tan \left (\mu x \right )^{m}d x}{a}+y \right )\right ) {\mathrm e}^{\frac {c \int _{}^{x}{\tan \left (\frac {\nu \left (b \int \tan \left (\lambda \textit {\_b} \right )^{-n} \tan \left (\mu \textit {\_b} \right )^{m}d \textit {\_b} -b \int \tan \left (\lambda x \right )^{-n} \tan \left (\mu x \right )^{m}d x +y a \right )}{a}\right )}^{k} \tan \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b}}{a}}\]
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