6.3.17 6.2

6.3.17.1 [941] Problem 1

problem number 941

Added Feb. 11, 2019.

Problem Chapter 3.6.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.17.2 [942] Problem 2

problem number 942

Added Feb. 11, 2019.

Problem Chapter 3.6.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.17.3 [943] Problem 3

problem number 943

Added Feb. 11, 2019.

Problem Chapter 3.6.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.17.4 [944] Problem 4

problem number 944

Added Feb. 11, 2019.

Problem Chapter 3.6.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.3.17.5 [945] Problem 5

problem number 945

Added Feb. 11, 2019.

Problem Chapter 3.6.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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