6.3.16 6.1

6.3.16.1 [936] Problem 1

problem number 936

Added Feb. 11, 2019.

Problem Chapter 3.6.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {1}{\mu \,ba\lambda } \left ( {\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \mu \,ba\lambda -ka\cos \left ( \mu \,y \right ) \lambda -\cos \left ( x\lambda \right ) c\mu \,b \right ) }\]

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6.3.16.2 [937] Problem 2

problem number 937

Added Feb. 11, 2019.

Problem Chapter 3.6.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.16.3 [938] Problem 3

problem number 938

Added Feb. 11, 2019.

Problem Chapter 3.6.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.16.4 [939] Problem 4

problem number 939

Added Feb. 11, 2019.

Problem Chapter 3.6.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.16.5 [940] Problem 5

problem number 940

Added Feb. 11, 2019.

Problem Chapter 3.6.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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