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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) {{\rm e}^{{\frac {c\sin \left ( x\lambda +\mu \,y \right ) }{a\lambda +\mu \,b}}}}\]

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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]];
 

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

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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) {{\rm e}^{{\frac {c\sin \left ( x\lambda \right ) \mu \,b+ak\lambda \,\sin \left ( \mu \,y \right ) }{a\lambda \,\mu \,b}}}}\]

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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]];
 

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

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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y}{x}} \right ) {{\rm e}^{{a\sin \left ( x\lambda +\mu \,y \right ) \left ( {\frac {\mu \,y}{x}}+\lambda \right ) ^{-1}}}}\]

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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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {b \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{n+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(\lambda x)\right )}{a \lambda n+a \lambda }+y\right ) \exp \left (\int _1^x\frac {s \cos ^k\left (\frac {\beta \left (b \csc (\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{n+1}(\lambda x)+a \lambda (n+1) y-b \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}\right )}{a \lambda (n+1)}\right )+c \cos ^m(\mu K[1])}{a}dK[1]\right )\right \}\right \}\]

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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac {b \left ( \cos \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( c \left ( \cos \left ( \mu \,{\it \_b} \right ) \right ) ^{m}+s \left ( \cos \left ( \beta \, \left ( -\int \!{\frac {b \left ( \cos \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}{\it \_b}+\int \!{\frac {b \left ( \cos \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x-y \right ) \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]

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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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {\sin ^2(\lambda y)} \csc (\lambda y) \cos ^{1-n}(\lambda y) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(\lambda y)\right )}{\lambda (n-1)}-\frac {b x}{a}\right ) \exp \left (\int _1^y\frac {\cos ^{-n}(\lambda K[1]) \left (s \cos ^k(\beta K[1])+c \cos ^m\left (\frac {\mu \left (a \csc (\lambda y) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(\lambda y)\right ) \sqrt {\sin ^2(\lambda y)} \cos ^{1-n}(\lambda y)-b \lambda (n-1) x-a \cos ^{1-n}(\lambda K[1]) \csc (\lambda K[1]) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}\right )}{b \lambda (n-1)}\right )\right )}{b}dK[1]\right )\right \}\right \}\]

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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {xb-a\int \! \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y}{b}} \right ) {{\rm e}^{\int ^{y}\!{\frac { \left ( \cos \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( c \left ( \cos \left ( {\frac {\mu }{b} \left ( a\int \! \left ( \cos \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y-\int \!{\frac { \left ( \cos \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}a}{b}}\,{\rm d}{\it \_b}b-xb \right ) } \right ) \right ) ^{m}+s \left ( \cos \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]

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