6.5.21 7.3

6.5.21.1 [1333] Problem 1

problem number 1333

Added April 13, 2019.

Problem Chapter 5.7.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.21.2 [1334] Problem 2

problem number 1334

Added April 13, 2019.

Problem Chapter 5.7.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.21.3 [1335] Problem 3

problem number 1335

Added April 13, 2019.

Problem Chapter 5.7.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.21.4 [1336] Problem 4

problem number 1336

Added April 13, 2019.

Problem Chapter 5.7.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.21.5 [1337] Problem 5

problem number 1337

Added April 13, 2019.

Problem Chapter 5.7.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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