6.5.7 4.1

6.5.7.1 [1249] Problem 1

problem number 1249

Added April 3, 2019.

Problem Chapter 5.4.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\sinh ^{k}\left (\mathit {\_a} \lambda \right )\right ) \left (\sinh ^{n}\left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta }{a}\right )\right ) {\mathrm e}^{-\frac {\mathit {\_a} c}{a}}}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]

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6.5.7.2 [1250] Problem 2

problem number 1250

Added April 3, 2019.

Problem Chapter 5.4.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \exp \left (\frac {c \sqrt {\cosh ^2(\lambda x)} \text {sech}(\lambda x) \sinh ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};-\sinh ^2(\lambda x)\right )}{a k \lambda +a \lambda }\right ) \left (\int _1^x\frac {\exp \left (-\frac {c \sqrt {\cosh ^2(\lambda K[1])} \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};-\sinh ^2(\lambda K[1])\right ) \text {sech}(\lambda K[1]) \sinh ^{k+1}(\lambda K[1])}{a \lambda +a k \lambda }\right ) s \sinh ^n(\beta K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \left (\int \frac {s \left (\sinh ^{n}\left (\beta x \right )\right ) {\mathrm e}^{-\frac {c \left (\int \left (\sinh ^{k}\left (\lambda x \right )\right )d x \right )}{a}}}{a}d x +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\int \frac {c \left (\sinh ^{k}\left (\lambda x \right )\right )}{a}d x}\]

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6.5.7.3 [1251] Problem 3

problem number 1251

Added April 3, 2019.

Problem Chapter 5.4.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \left (c_1 \sinh ^{n_1}(\lambda _1 x)+ c_2 \sinh ^{n_2}(\lambda _2 y) \right ) w + s_1 \sinh ^{k_1}(\beta _1 x)+ s_2 \sinh ^{k_2}(\beta _2 y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c1*Sinh[lambda1*x]^n1 + c2*Sinh[lambda2*y]^n2)*w[x,y] + s1*Sinh[beta1*x]^k1+ s2*Sinh[beta2*y]^k2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\frac {\text {c1} \sqrt {\cosh ^2(\text {lambda1} x)} \text {sech}(\text {lambda1} x) \sinh ^{\text {n1}+1}(\text {lambda1} x) \, _2F_1\left (\frac {1}{2},\frac {\text {n1}+1}{2};\frac {\text {n1}+3}{2};-\sinh ^2(\text {lambda1} x)\right )}{a \text {lambda1} \text {n1}+a \text {lambda1}}+\frac {\text {c2} \sqrt {\cosh ^2(\text {lambda2} y)} \text {sech}(\text {lambda2} y) \sinh ^{\text {n2}+1}(\text {lambda2} y) \, _2F_1\left (\frac {1}{2},\frac {\text {n2}+1}{2};\frac {\text {n2}+3}{2};-\sinh ^2(\text {lambda2} y)\right )}{b \text {lambda2} \text {n2}+b \text {lambda2}}\right ) \left (\int _1^x\frac {\exp \left (-\frac {\text {c1} \sqrt {\cosh ^2(\text {lambda1} K[1])} \, _2F_1\left (\frac {1}{2},\frac {\text {n1}+1}{2};\frac {\text {n1}+3}{2};-\sinh ^2(\text {lambda1} K[1])\right ) \text {sech}(\text {lambda1} K[1]) \sinh ^{\text {n1}+1}(\text {lambda1} K[1])}{a \text {lambda1}+a \text {n1} \text {lambda1}}-\frac {\text {c2} \sqrt {\cosh ^2\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )} \, _2F_1\left (\frac {1}{2},\frac {\text {n2}+1}{2};\frac {\text {n2}+3}{2};-\sinh ^2\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right ) \text {sech}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) \sinh ^{\text {n2}+1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{b \text {lambda2}+b \text {n2} \text {lambda2}}\right ) \left (\text {s1} \sinh ^{\text {k1}}(\text {beta1} K[1])+\text {s2} \sinh ^{\text {k2}}\left (\text {beta2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c1*sinh(lambda1*x)^n1 + c2*sinh(lambda2*y)^n2)*w(x,y) + s1*sinh(beta1*x)^k1+ s2*sinh(beta2*y)^k2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {s1} \left (\sinh ^{\mathit {k1}}\left (\mathit {\_b} \beta 1 \right )\right )+\mathit {s2} \left (\sinh ^{\mathit {k2}}\left (\frac {\left (a y -\left (-\mathit {\_b} +x \right ) b \right ) \beta 2}{a}\right )\right )\right ) {\mathrm e}^{-\frac {\int \left (\mathit {c1} \left (\sinh ^{\mathit {n1}}\left (\mathit {\_b} \lambda 1 \right )\right )+\mathit {c2} \left (\sinh ^{\mathit {n2}}\left (\frac {\left (a y -\left (-\mathit {\_b} +x \right ) b \right ) \lambda 2}{a}\right )\right )\right )d \mathit {\_b}}{a}}}{a}d\mathit {\_b} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {\mathit {c1} \left (\sinh ^{\mathit {n1}}\left (\mathit {\_a} \lambda 1 \right )\right )+\mathit {c2} \left (\sinh ^{\mathit {n2}}\left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \lambda 2}{a}\right )\right )}{a}d\mathit {\_a}}\]

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6.5.7.4 [1252] Problem 4

problem number 1252

Added April 3, 2019.

Problem Chapter 5.4.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \exp \left (\frac {c 2^{n-1} e^{-\nu x} \left (e^{\lambda x}-e^{-\lambda x}\right )^{-n} \left (1-e^{2 \lambda x}\right )^n \left ((\lambda n+\nu ) \, _2F_1\left (n,\frac {\lambda n-\nu }{2 \lambda };\frac {1}{2} \left (n-\frac {\nu }{\lambda }+2\right );e^{2 \lambda x}\right )+e^{2 \nu x} (\nu -\lambda n) \, _2F_1\left (n,\frac {\lambda n+\nu }{2 \lambda };\frac {\lambda n+\nu }{2 \lambda }+1;e^{2 \lambda x}\right )\right )}{a (\nu -\lambda n) (\lambda n+\nu )}\right ) \left (\int _1^x\frac {\exp \left (-\frac {2^{n-1} c e^{-\nu K[2]} \left (-e^{-\lambda K[2]}+e^{\lambda K[2]}\right )^{-n} \left (1-e^{2 \lambda K[2]}\right )^n \left ((\lambda n+\nu ) \, _2F_1\left (n,\frac {\lambda n-\nu }{2 \lambda };\frac {1}{2} \left (n-\frac {\nu }{\lambda }+2\right );e^{2 \lambda K[2]}\right )+e^{2 \nu K[2]} (\nu -\lambda n) \, _2F_1\left (n,\frac {\lambda n+\nu }{2 \lambda };\frac {\lambda n+\nu }{2 \lambda }+1;e^{2 \lambda K[2]}\right )\right )}{a (\nu -\lambda n) (\lambda n+\nu )}\right ) p \sinh ^{-n}(\lambda K[2]) \sinh ^s\left (\beta \left (y-\int _1^x\frac {b \sinh ^{-n}(\lambda K[1]) \sinh ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \sinh ^{-n}(\lambda K[1]) \sinh ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]+c_1\left (y-\int _1^x\frac {b \sinh ^{-n}(\lambda K[1]) \sinh ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]

Maple ✓

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

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

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6.5.7.5 [1253] Problem 5

problem number 1253

Added April 3, 2019.

Problem Chapter 5.4.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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