6.5.10 4.4

6.5.10.1 [1265] Problem 1

problem number 1265

Added April 3, 2019.

Problem Chapter 5.4.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.10.2 [1266] Problem 2

problem number 1266

Added April 3, 2019.

Problem Chapter 5.4.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \coth ^k(\lambda x) w + s \coth ^n(\beta x) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \exp \left (\frac {c \coth ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},\coth ^2(\lambda x)\right )}{a k \lambda +a \lambda }\right ) \left (\int _1^x\frac {\exp \left (-\frac {c \coth ^{k+1}(\lambda K[1]) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},\coth ^2(\lambda K[1])\right )}{a \lambda +a k \lambda }\right ) s \coth ^n(\beta K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

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

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

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6.5.10.3 [1267] Problem 3

problem number 1267

Added April 3, 2019.

Problem Chapter 5.4.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \left (c_1 \coth ^{n_1}(\lambda _1 x)+ c_2 \coth ^{n_2}(\lambda _2 y) \right ) w + s_1 \coth ^{k_1}(\beta _1 x)+ s_2 \coth ^{k_2}(\beta _2 y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == (c1*Coth[lambda1*x]^n1 + c2*Coth[lambda2*y]^n2)*w[x,y] + s1*Coth[beta1*x]^k1+ s2*Coth[beta2*y]^k2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\frac {\text {c1} \coth ^{\text {n1}+1}(\text {lambda1} x) \operatorname {Hypergeometric2F1}\left (1,\frac {\text {n1}+1}{2},\frac {\text {n1}+3}{2},\coth ^2(\text {lambda1} x)\right )}{a \text {lambda1} \text {n1}+a \text {lambda1}}+\frac {\text {c2} \coth ^{\text {n2}+1}(\text {lambda2} y) \operatorname {Hypergeometric2F1}\left (1,\frac {\text {n2}+1}{2},\frac {\text {n2}+3}{2},\coth ^2(\text {lambda2} y)\right )}{b \text {lambda2} \text {n2}+b \text {lambda2}}\right ) \left (\int _1^x\frac {\exp \left (-\frac {\text {c1} \operatorname {Hypergeometric2F1}\left (1,\frac {\text {n1}+1}{2},\frac {\text {n1}+3}{2},\coth ^2(\text {lambda1} K[1])\right ) \coth ^{\text {n1}+1}(\text {lambda1} K[1])}{a \text {lambda1}+a \text {n1} \text {lambda1}}-\frac {\text {c2} \coth ^{\text {n2}+1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {\text {n2}+1}{2},\frac {\text {n2}+3}{2},\coth ^2\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right )}{b \text {lambda2}+b \text {n2} \text {lambda2}}\right ) \left (\text {s1} \coth ^{\text {k1}}(\text {beta1} K[1])+\text {s2} \coth ^{\text {k2}}\left (\text {beta2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c1*coth(lambda1*x)^n1 + c2*coth(lambda2*y)^n2)*w(x,y) + s1*coth(beta1*x)^k1+ s2*coth(beta2*y)^k2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}{\mathrm e}^{-\frac {\int \left (\operatorname {c1} \coth \left (\lambda \operatorname {1} \textit {\_a} \right )^{\operatorname {n1}}+\operatorname {c2} \coth \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \lambda \operatorname {2} }{a}\right )^{\operatorname {n2}}\right )d \textit {\_a}}{a}} \left (\operatorname {s1} \coth \left (\beta \operatorname {1} \textit {\_a} \right )^{\operatorname {k1}}+\operatorname {s2} \coth \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \beta \operatorname {2} }{a}\right )^{\operatorname {k2}}\right )d \textit {\_a}}{a}+f_{1} \left (y -\frac {b x}{a}\right )\right ) {\mathrm e}^{\frac {\int _{}^{x}\left (\operatorname {c1} \coth \left (\lambda \operatorname {1} \textit {\_a} \right )^{\operatorname {n1}}+\operatorname {c2} \coth \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \lambda \operatorname {2} }{a}\right )^{\operatorname {n2}}\right )d \textit {\_a}}{a}}\]

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6.5.10.4 [1268] Problem 4

problem number 1268

Added April 3, 2019.

Problem Chapter 5.4.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \coth ^n(\lambda x) w_x + b \coth ^m(\mu x) w_y = c \coth ^k(\nu x) w + p \coth ^s(\beta y) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \coth ^{-n}(\lambda K[2]) \coth (\nu K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \coth ^{-n}(\lambda K[2]) \coth (\nu K[2])}{a}dK[2]\right ) p \coth ^{-n}(\lambda K[3]) \coth ^s\left (\beta \left (y-\int _1^x\frac {b \coth ^{-n}(\lambda K[1]) \coth ^m(\mu K[1])}{a}dK[1]+\int _1^{K[3]}\frac {b \coth ^{-n}(\lambda K[1]) \coth ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \coth ^{-n}(\lambda K[1]) \coth ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]

Maple ✓

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

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

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6.5.10.5 [1269] Problem 5

problem number 1269

Added April 3, 2019.

Problem Chapter 5.4.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \coth ^n(\lambda x) w_x + b \coth ^m(\mu x) w_y = c \coth ^k(\nu y) w + p \coth ^s(\beta x) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \coth ^{-n}(\lambda K[2]) \coth \left (\nu \left (y-\int _1^x\frac {b \coth ^{-n}(\lambda K[1]) \coth ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \coth ^{-n}(\lambda K[1]) \coth ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \coth ^{-n}(\lambda K[2]) \coth \left (\nu \left (y-\int _1^x\frac {b \coth ^{-n}(\lambda K[1]) \coth ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \coth ^{-n}(\lambda K[1]) \coth ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) p \coth ^s(\beta K[3]) \coth ^{-n}(\lambda K[3])}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \coth ^{-n}(\lambda K[1]) \coth ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]

Maple ✓

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

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

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