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'));
 

\[w \left (x , y\right ) = \frac {a b \lambda \mu \mathit {\_F1} \left (\frac {y a -b x}{a}\right )-a k \lambda \cos \left (\mu y \right )-b c \mu \cos \left (\lambda x \right )}{a b \lambda \mu }\]

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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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

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

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