6.5.15 6.2

6.5.15.1 [1296] Problem 1

problem number 1296

Added April 11, 2019.

Problem Chapter 5.6.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = {\mathrm e}^{\frac {c x}{a}} f_{1} \left (\frac {y a -b x}{a}\right )+\frac {\left (-c \cos \left (\lambda x +\mu y \right )+\left (a \lambda +b \mu \right ) \sin \left (\lambda x +\mu y \right )\right ) k}{a^{2} \lambda ^{2}+2 a b \lambda \mu +b^{2} \mu ^{2}+c^{2}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.5.15.2 [1297] Problem 2

problem number 1297

Added April 11, 2019.

Problem Chapter 5.6.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = w + c_1 \cos ^k(\lambda x) + c_2 \cos ^n(\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*Cos[lambda*x]^k + c2*Cos[beta*y]^n; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \frac {e^{\frac {x}{a}} (a k \lambda -i) (b \beta n-i) c_1\left (y-\frac {b x}{a}\right )+\text {c1} 2^{-k} (1+i b \beta n) \left (e^{-i \lambda x}+e^{i \lambda x}\right )^k \left (1+e^{2 i \lambda x}\right )^{-k} \operatorname {Hypergeometric2F1}\left (-k,\frac {i}{2 a \lambda }-\frac {k}{2},-\frac {k}{2}+\frac {i}{2 a \lambda }+1,-e^{2 i \lambda x}\right )+\text {c2} 2^n (1+i a k \lambda ) \cos ^n(\beta y) (\cosh (n \log (2))-\sinh (n \log (2))) (i \sin (2 \beta y)+\cos (2 \beta y)+1)^{-n} \operatorname {Hypergeometric2F1}\left (\frac {i}{2 b \beta }-\frac {n}{2},-n,-\frac {n}{2}+\frac {i}{2 b \beta }+1,-\cos (2 \beta y)-i \sin (2 \beta y)\right )}{(a k \lambda -i) (b \beta n-i)}\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+ c1*cos(lambda*x)^k + c2*cos(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}\left (\operatorname {c1} \cos \left (\lambda \textit {\_a} \right )^{k}+\operatorname {c2} \cos \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \beta }{a}\right )^{n}\right ) {\mathrm e}^{-\frac {\textit {\_a}}{a}}d \textit {\_a}}{a}+f_{1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.5.15.3 [1298] Problem 3

problem number 1298

Added April 11, 2019.

Problem Chapter 5.6.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \cos ^k(\lambda K[1]) \cos ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.5.15.4 [1299] Problem 4

problem number 1299

Added April 11, 2019.

Problem Chapter 5.6.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.5.15.5 [1300] Problem 5

problem number 1300

Added April 11, 2019.

Problem Chapter 5.6.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^xa \cos \left (\left (\lambda +\frac {\mu y}{x}\right ) K[1]\right )dK[1]\right ) \left (\int _1^x\frac {b \exp \left (-\int _1^{K[2]}a \cos \left (\left (\lambda +\frac {\mu y}{x}\right ) K[1]\right )dK[1]\right ) \cos (\nu K[2])}{K[2]}dK[2]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \left (b \int _{}^{x}\frac {\cos \left (\nu \textit {\_a} \right ) {\mathrm e}^{-\frac {a \sin \left (\frac {\textit {\_a} \left (\lambda x +y \mu \right )}{x}\right ) x}{\lambda x +y \mu }}}{\textit {\_a}}d \textit {\_a} +f_{1} \left (\frac {y}{x}\right )\right ) {\mathrm e}^{\frac {a \sin \left (\lambda x +y \mu \right ) x}{\lambda x +y \mu }}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.5.15.6 [1301] Problem 6

problem number 1301

Added April 11, 2019.

Problem Chapter 5.6.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________