6.4.16 6.2

6.4.16.1 [1119] Problem 1

problem number 1119

Added March 9, 2019.

Problem Chapter 4.6.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.16.2 [1120] Problem 2

problem number 1120

Added March 9, 2019.

Problem Chapter 4.6.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.16.3 [1121] Problem 3

problem number 1121

Added March 9, 2019.

Problem Chapter 4.6.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a x \cos (\lambda x+ \mu y) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right ) \exp \left (\int _1^xa \cos \left (\left (\lambda +\frac {\mu y}{x}\right ) K[1]\right )dK[1]\right )\right \}\right \}\]

Maple ✓

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

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

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6.4.16.4 [1122] Problem 4

problem number 1122

Added March 9, 2019.

Problem Chapter 4.6.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.16.5 [1123] Problem 5

problem number 1123

Added March 9, 2019.

Problem Chapter 4.6.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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