6.5.14 6.1

6.5.14.1 [1289] Problem 1

problem number 1289

Added April 8, 2019.

Problem Chapter 5.6.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c w + k \sin (\lambda x+\mu y) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \frac {k e^{-i (\lambda x+\mu y)} \left (i (a \lambda +b \mu ) \left (1+e^{2 i (\lambda x+\mu y)}\right )+c \left (-1+e^{2 i (\lambda x+\mu y)}\right )\right )-2 i e^{\frac {c x}{a}} \left ((a \lambda +b \mu )^2+c^2\right ) c_1\left (y-\frac {b x}{a}\right )}{2 (a \lambda +b \mu -i c) (c-i (a \lambda +b \mu ))}\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = {\mathrm e}^{\frac {c x}{a}} f_{1} \left (\frac {y a -b x}{a}\right )-\frac {k \left (\left (a \lambda +b \mu \right ) \cos \left (\lambda x +y \mu \right )+c \sin \left (\lambda x +y \mu \right )\right )}{a^{2} \lambda ^{2}+2 a b \lambda \mu +b^{2} \mu ^{2}+c^{2}}\]

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6.5.14.2 [1290] Problem 2

problem number 1290

Added April 8, 2019.

Problem Chapter 5.6.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \frac {e^{\frac {x}{a}} (a k \lambda -i) (b \beta n-i) c_1\left (y-\frac {b x}{a}\right )+\text {c1} (1+i b \beta n) \left (-i e^{-i \lambda x} \left (-1+e^{2 i \lambda x}\right )\right )^k \left (2-2 e^{2 i \lambda x}\right )^{-k} \operatorname {Hypergeometric2F1}\left (-k,\frac {i}{2 a \lambda }-\frac {k}{2},-\frac {k}{2}+\frac {i}{2 a \lambda }+1,e^{2 i \lambda x}\right )+\text {c2} 2^{-n} (1+i a k \lambda ) \left (-1+e^{2 i \beta y}\right )^n \left (-i e^{-i \beta y} \left (-1+e^{2 i \beta y}\right )\right )^n \left (-\left (-1+e^{2 i \beta y}\right )^2\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {i}{2 b \beta }-\frac {n}{2},-n,-\frac {n}{2}+\frac {i}{2 b \beta }+1,e^{2 i \beta y}\right )}{(a k \lambda -i) (b \beta n-i)}\right \}\right \}\]

Maple ✓

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

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

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6.5.14.3 [1291] Problem 3

problem number 1291

Added April 8, 2019.

Problem Chapter 5.6.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.14.4 [1292] Problem 4

problem number 1292

Added April 8, 2019.

Problem Chapter 5.6.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = c w +k \sin (\lambda x+\mu y) \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.14.5 [1293] Problem 5

problem number 1293

Added April 8, 2019.

Problem Chapter 5.6.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a x \sin (\lambda x+\mu y) w + b \sin (\nu x) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^xa \sin \left (\left (\lambda +\frac {\beta y}{x}\right ) K[1]\right )dK[1]\right ) \left (\int _1^x\frac {b \exp \left (-\int _1^{K[2]}a \sin \left (\left (\lambda +\frac {\beta y}{x}\right ) K[1]\right )dK[1]\right ) \sin (\nu K[2])}{K[2]}dK[2]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]

Maple ✓

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

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

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6.5.14.6 [1294] Problem 6

problem number 1294

Added April 8, 2019.

Problem Chapter 5.6.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.14.7 [1295] Problem 7

problem number 1295

Added April 8, 2019.

Problem Chapter 5.6.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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