6.5.22 7.4

6.5.22.1 [1338] Problem 1

problem number 1338

Added April 13, 2019.

Problem Chapter 5.7.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = w + c_1 \arccot ^k(\lambda x) + c_2 \arccot ^n(\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*ArcCot[lambda*x]^k+c2*ArcCot[beta*y]^n; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} \cot ^{-1}(\lambda K[1])^k+\text {c2} \cot ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) =  w(x,y)+c1*arccot(lambda*x)^k+c2*arccot(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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6.5.22.2 [1339] Problem 2

problem number 1339

Added April 13, 2019.

Problem Chapter 5.7.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.22.3 [1340] Problem 3

problem number 1340

Added April 13, 2019.

Problem Chapter 5.7.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \left ( c_1 \arccot (\lambda _1 x) + c_2 \arccot (\lambda _2 y)\right ) w+ s_1 \arccot ^n(\beta _1 x)+ s_2 \arccot ^k(\beta _2 y) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {\text {c1} \cot ^{-1}(\text {lambda1} K[1])+\text {c2} \cot ^{-1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {c1} \cot ^{-1}(\text {lambda1} K[1])+\text {c2} \cot ^{-1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]\right ) \left (\text {s2} \cot ^{-1}\left (\text {beta2} \left (y+\frac {b (K[2]-x)}{a}\right )\right )^k+\text {s1} \cot ^{-1}(\text {beta1} K[2])^n\right )}{a}dK[2]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

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

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

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6.5.22.4 [1341] Problem 4

problem number 1341

Added April 13, 2019.

Problem Chapter 5.7.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.22.5 [1342] Problem 5

problem number 1342

Added April 13, 2019.

Problem Chapter 5.7.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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