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]];
 

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

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'));
 

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

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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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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