6.5.18 6.5
6.5.18.1 [1316] Problem 1
problem number 1316
Added April 11, 2019.
Problem Chapter 5.6.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = w + c_1 \sin ^k(\lambda x)+c_2 \cos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+c1*Sin[lambda*x]^k+c2*Cos[beta*y]^n;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {e^{\frac {x}{a}} (a k \lambda -i) (b \beta n-i) c_1\left (y-\frac {b x}{a}\right )+\text {c1} (1+i b \beta n) \left (-i e^{-i \lambda x} \left (-1+e^{2 i \lambda x}\right )\right )^k \left (2-2 e^{2 i \lambda x}\right )^{-k} \operatorname {Hypergeometric2F1}\left (-k,\frac {i}{2 a \lambda }-\frac {k}{2},-\frac {k}{2}+\frac {i}{2 a \lambda }+1,e^{2 i \lambda x}\right )+\text {c2} 2^n (1+i a k \lambda ) \cos ^n(\beta y) (\cosh (n \log (2))-\sinh (n \log (2))) (i \sin (2 \beta y)+\cos (2 \beta y)+1)^{-n} \operatorname {Hypergeometric2F1}\left (\frac {i}{2 b \beta }-\frac {n}{2},-n,-\frac {n}{2}+\frac {i}{2 b \beta }+1,-\cos (2 \beta y)-i \sin (2 \beta y)\right )}{(a k \lambda -i) (b \beta n-i)}\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*sin(lambda*x)^k+c2*cos(beta*y)^n;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}\left (\operatorname {c1} \sin \left (\lambda \textit {\_a} \right )^{k}+\operatorname {c2} \cos \left (\frac {\beta \left (a y -b \left (x -\textit {\_a} \right )\right )}{a}\right )^{n}\right ) {\mathrm e}^{-\frac {\textit {\_a}}{a}}d \textit {\_a}}{a}+f_{1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]
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6.5.18.2 [1317] Problem 2
problem number 1317
Added April 11, 2019.
Problem Chapter 5.6.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \sin ^k(\lambda x) \cos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+Sin[lambda*x]^k*Cos[beta*y]^n;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \cos ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right ) \sin ^k(\lambda K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+sin(lambda*x)^k*cos(beta*y)^n;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}\sin \left (\lambda \textit {\_a} \right )^{k} \cos \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \beta }{a}\right )^{n} {\mathrm e}^{-\frac {c \textit {\_a}}{a}}d \textit {\_a}}{a}+f_{1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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6.5.18.3 [1318] Problem 3
problem number 1318
Added April 11, 2019.
Problem Chapter 5.6.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \sin (\mu y) w_y = c \sin (\lambda x) w + k \cos (\nu x) + s \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*Sin[mu*y]*D[w[x, y], y] == c*Sin[lambda*x]*w[x,y]+k*Cos[nu*x]+s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\begin{align*}& \left \{w(x,y)\to \exp \left (\int _1^x\frac {c \sin (\lambda K[1])}{a}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \sin (\lambda K[1])}{a}dK[1]\right ) (s+k \cos (\nu K[3]))}{a}dK[3]+c_1\left (-\frac {b x}{a}-\frac {\text {arctanh}(\cos (\mu y))}{\mu }\right )\right )\right \}\\& \left \{w(x,y)\to \exp \left (\int _1^x\frac {c \sin (\lambda K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[4]}\frac {c \sin (\lambda K[2])}{a}dK[2]\right ) (s+k \cos (\nu K[4]))}{a}dK[4]+c_1\left (-\frac {b x}{a}-\frac {\text {arctanh}(\cos (\mu y))}{\mu }\right )\right )\right \}\\\end{align*}
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*sin(mu*y)*diff(w(x,y),y) = c*sin(lambda*x)*w(x,y)+k*cos(nu*x)+s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{y}\left (k \cos \left (\frac {\nu \left (x b \mu +a \ln \left (\csc \left (\mu y \right )+\cot \left (\mu y \right )\right )-a \ln \left (\csc \left (\mu \textit {\_a} \right )+\cot \left (\mu \textit {\_a} \right )\right )\right )}{b \mu }\right )+s \right ) {\mathrm e}^{\frac {c \cos \left (\frac {\lambda \left (x b \mu +a \ln \left (\csc \left (\mu y \right )+\cot \left (\mu y \right )\right )-a \ln \left (\csc \left (\mu \textit {\_a} \right )+\cot \left (\mu \textit {\_a} \right )\right )\right )}{b \mu }\right )}{a \lambda }} \csc \left (\mu \textit {\_a} \right )d \textit {\_a}}{b}+f_{1} \left (\frac {x b \mu +a \ln \left (\csc \left (\mu y \right )+\cot \left (\mu y \right )\right )}{b \mu }\right )\right ) {\mathrm e}^{-\frac {c \cos \left (\lambda x \right )}{a \lambda }}\]
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6.5.18.4 [1319] Problem 4
problem number 1319
Added April 11, 2019.
Problem Chapter 5.6.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \sin (\mu y) w_y = c \sin (\lambda x) w + k \tan (\nu x) + s \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*Sin[mu*y]*D[w[x, y], y] == c*Sin[lambda*x]*w[x,y]+k*Tan[nu*x]+s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\begin{align*}& \left \{w(x,y)\to \exp \left (\int _1^x\frac {c \sin (\lambda K[1])}{a}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \sin (\lambda K[1])}{a}dK[1]\right ) (s+k \tan (\nu K[3]))}{a}dK[3]+c_1\left (-\frac {b x}{a}-\frac {\text {arctanh}(\cos (\mu y))}{\mu }\right )\right )\right \}\\& \left \{w(x,y)\to \exp \left (\int _1^x\frac {c \sin (\lambda K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[4]}\frac {c \sin (\lambda K[2])}{a}dK[2]\right ) (s+k \tan (\nu K[4]))}{a}dK[4]+c_1\left (-\frac {b x}{a}-\frac {\text {arctanh}(\cos (\mu y))}{\mu }\right )\right )\right \}\\\end{align*}
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*sin(mu*y)*diff(w(x,y),y) = c*sin(lambda*x)*w(x,y)+k*tan(nu*x)+s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{y}{\mathrm e}^{\frac {c \cos \left (\frac {\lambda \left (x b \mu +a \ln \left (\csc \left (\mu y \right )+\cot \left (\mu y \right )\right )-a \ln \left (\csc \left (\mu \textit {\_a} \right )+\cot \left (\mu \textit {\_a} \right )\right )\right )}{b \mu }\right )}{a \lambda }} \csc \left (\mu \textit {\_a} \right ) \left (k \tan \left (\frac {\nu \left (x b \mu +a \ln \left (\csc \left (\mu y \right )+\cot \left (\mu y \right )\right )-a \ln \left (\csc \left (\mu \textit {\_a} \right )+\cot \left (\mu \textit {\_a} \right )\right )\right )}{b \mu }\right )+s \right )d \textit {\_a}}{b}+f_{1} \left (\frac {x b \mu +a \ln \left (\csc \left (\mu y \right )+\cot \left (\mu y \right )\right )}{b \mu }\right )\right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right ) c}{a \lambda }}\]
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6.5.18.5 [1320] Problem 5
problem number 1320
Added April 11, 2019.
Problem Chapter 5.6.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \tan (\mu y) w_y = c \tan (\lambda x) w + k \cot (\nu x) + s \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*Tan[mu*y]*D[w[x, y], y] == c*Tan[lambda*x]*w[x,y]+k*Cot[nu*x]+s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \cos ^{-\frac {c}{a \lambda }}(\lambda x) \left (\int _1^x\frac {\cos ^{\frac {c}{a \lambda }}(\lambda K[1]) (s+k \cot (\nu K[1]))}{a}dK[1]+c_1\left (\frac {\log (\sin (\mu y))}{\mu }-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*tan(mu*y)*diff(w(x,y),y) = c*tan(lambda*x)*w(x,y)+k*cot(nu*x)+s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int \left (k \cot \left (\nu x \right )+s \right ) \left (\sec \left (\lambda x \right )^{2}\right )^{-\frac {c}{2 a \lambda }}d x}{a}+f_{1} \left (-x +\frac {\ln \left (\operatorname {csgn}\left (\sec \left (\mu y \right )\right ) \sin \left (\mu y \right )\right ) a}{b \mu }\right )\right ) \left (\sec \left (\lambda x \right )^{2}\right )^{\frac {c}{2 a \lambda }}\]
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6.5.18.6 [1321] Problem 6
problem number 1321
Added April 11, 2019.
Problem Chapter 5.6.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \sin ^n(\lambda x) w_x + b \cos ^m(\mu x) w_y = c \cos ^k(\nu x) w + p \sin ^s(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*Sin[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*Sin[beta*y]^s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \cos ^k(\nu K[2]) \sin ^{-n}(\lambda K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \cos ^k(\nu K[2]) \sin ^{-n}(\lambda K[2])}{a}dK[2]\right ) p \sin ^{-n}(\lambda K[3]) \sin ^s\left (\beta \left (y-\int _1^x\frac {b \cos ^m(\mu K[1]) \sin ^{-n}(\lambda K[1])}{a}dK[1]+\int _1^{K[3]}\frac {b \cos ^m(\mu K[1]) \sin ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \cos ^m(\mu K[1]) \sin ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*sin(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*sin(beta*y)^s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {p \int _{}^{x}{\sin \left (\frac {\beta \left (b \int \cos \left (\mu \textit {\_f} \right )^{m} \sin \left (\lambda \textit {\_f} \right )^{-n}d \textit {\_f} -b \int \cos \left (\mu x \right )^{m} \sin \left (\lambda x \right )^{-n}d x +y a \right )}{a}\right )}^{s} \sin \left (\lambda \textit {\_f} \right )^{-n} {\mathrm e}^{-\frac {c \int \cos \left (\nu \textit {\_f} \right )^{k} \sin \left (\lambda \textit {\_f} \right )^{-n}d \textit {\_f}}{a}}d \textit {\_f}}{a}+f_{1} \left (-\frac {b \int \cos \left (\mu x \right )^{m} \sin \left (\lambda x \right )^{-n}d x}{a}+y \right )\right ) {\mathrm e}^{\frac {c \int \cos \left (\nu x \right )^{k} \sin \left (\lambda x \right )^{-n}d x}{a}}\]
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6.5.18.7 [1322] Problem 7
problem number 1322
Added April 11, 2019.
Problem Chapter 5.6.5.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \tan ^n(\lambda x) w_x + b \cot ^m(\mu x) w_y = c \tan ^k(\nu x) w + p \cot ^s(\beta x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*Tan[lambda*x]^n*D[w[x, y], x] + b*Cot[mu*x]^m*D[w[x, y], y] == c*Tan[nu*x]^k*w[x,y]+p*Cot[beta*x]^s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k(\nu K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k(\nu K[2])}{a}dK[2]\right ) p \cot ^s(\beta K[3]) \tan ^{-n}(\lambda K[3])}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \cot ^m(\mu K[1]) \tan ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*tan(lambda*x)^n*diff(w(x,y),x)+ b*cot(mu*x)^m*diff(w(x,y),y) = c*tan(nu*x)^k*w(x,y)+p*cot(beta*x)^s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {p \int \cot \left (\beta x \right )^{s} \tan \left (\lambda x \right )^{-n} {\mathrm e}^{-\frac {c \int \tan \left (\nu x \right )^{k} \tan \left (\lambda x \right )^{-n}d x}{a}}d x}{a}+f_{1} \left (-\frac {b \int \cot \left (\mu x \right )^{m} \tan \left (\lambda x \right )^{-n}d x}{a}+y \right )\right ) {\mathrm e}^{\frac {c \int \tan \left (\nu x \right )^{k} \tan \left (\lambda x \right )^{-n}d x}{a}}\]
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