6.5.8 4.2
6.5.8.1 [1254] Problem 1
problem number 1254
Added April 3, 2019.
Problem Chapter 5.4.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \cosh ^k(\lambda x) \cosh ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+Cosh[lambda*x]^k*Cosh[beta*y]^n;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \cosh ^k(\lambda K[1]) \cosh ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+cosh(lambda*x)^k*cosh(beta*y)^n;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}\cosh \left (\lambda \textit {\_a} \right )^{k} \cosh \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \beta }{a}\right )^{n} {\mathrm e}^{-\frac {c \textit {\_a}}{a}}d \textit {\_a}}{a}+f_{1} \left (\frac {a y -x b}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.5.8.2 [1255] Problem 2
problem number 1255
Added April 3, 2019.
Problem Chapter 5.4.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \cosh ^k(\lambda x) w + s \cosh ^n(\beta x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Cosh[lambda*x]^k*w[x,y]+ s*Cosh[beta*x]^n;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {c \sqrt {-\sinh ^2(\lambda x)} \text {csch}(\lambda x) \cosh ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\cosh ^2(\lambda x)\right )}{a k \lambda +a \lambda }\right ) \left (\int _1^x\frac {\exp \left (\frac {c \cosh ^{k+1}(\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\cosh ^2(\lambda K[1])\right ) \sinh (\lambda K[1])}{(a \lambda +a k \lambda ) \sqrt {-\sinh ^2(\lambda K[1])}}\right ) s \cosh ^n(\beta K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*cosh(lambda*x)^k*w(x,y)+s*cosh(beta*x)^n;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {s \int \cosh \left (\beta x \right )^{n} {\mathrm e}^{-\frac {c \int \cosh \left (\lambda x \right )^{k}d x}{a}}d x}{a}+f_{1} \left (y -\frac {b x}{a}\right )\right ) {\mathrm e}^{\frac {c \int \cosh \left (\lambda x \right )^{k}d x}{a}}\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.5.8.3 [1256] Problem 3
problem number 1256
Added April 3, 2019.
Problem Chapter 5.4.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = \left (c_1 \cosh ^{n_1}(\lambda _1 x)+ c_2 \cosh ^{n_2}(\lambda _2 y) \right ) w + s_1 \cosh ^{k_1}(\beta _1 x)+ s_2 \cosh ^{k_2}(\beta _2 y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c1*Cosh[lambda1*x]^n1 + c2*Cosh[lambda2*y]^n2)*w[x,y] + s1*Cosh[beta1*x]^k1+ s2*Cosh[beta2*y]^k2;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {\text {c1} \sqrt {-\sinh ^2(\text {lambda1} x)} \text {csch}(\text {lambda1} x) \cosh ^{\text {n1}+1}(\text {lambda1} x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {\text {n1}+1}{2},\frac {\text {n1}+3}{2},\cosh ^2(\text {lambda1} x)\right )}{a \text {lambda1} \text {n1}+a \text {lambda1}}+\frac {\text {c2} \sqrt {-\sinh ^2(\text {lambda2} y)} \text {csch}(\text {lambda2} y) \cosh ^{\text {n2}+1}(\text {lambda2} y) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {\text {n2}+1}{2},\frac {\text {n2}+3}{2},\cosh ^2(\text {lambda2} y)\right )}{b \text {lambda2} \text {n2}+b \text {lambda2}}\right ) \left (\int _1^x\frac {\exp \left (\frac {\text {c1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {\text {n1}+1}{2},\frac {\text {n1}+3}{2},\cosh ^2(\text {lambda1} K[1])\right ) \sinh (\text {lambda1} K[1]) \cosh ^{\text {n1}+1}(\text {lambda1} K[1])}{(a \text {lambda1}+a \text {n1} \text {lambda1}) \sqrt {-\sinh ^2(\text {lambda1} K[1])}}+\frac {\text {c2} \cosh ^{\text {n2}+1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {\text {n2}+1}{2},\frac {\text {n2}+3}{2},\cosh ^2\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right ) \sinh \left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{(b \text {lambda2}+b \text {n2} \text {lambda2}) \sqrt {-\sinh ^2\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )}}\right ) \left (\text {s1} \cosh ^{\text {k1}}(\text {beta1} K[1])+\text {s2} \cosh ^{\text {k2}}\left (\text {beta2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c1*cosh(lambda1*x)^n1 + c2*cosh(lambda2*y)^n2)*w(x,y) + s1*cosh(beta1*x)^k1+ s2*cosh(beta2*y)^k2;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}\left (\operatorname {s1} \cosh \left (\beta \operatorname {1} \textit {\_a} \right )^{\operatorname {k1}}+\operatorname {s2} \cosh \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \beta \operatorname {2} }{a}\right )^{\operatorname {k2}}\right ) {\mathrm e}^{-\frac {\int \left (\operatorname {c1} \cosh \left (\lambda \operatorname {1} \textit {\_a} \right )^{\operatorname {n1}}+\operatorname {c2} \cosh \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \lambda \operatorname {2} }{a}\right )^{\operatorname {n2}}\right )d \textit {\_a}}{a}}d \textit {\_a}}{a}+f_{1} \left (y -\frac {b x}{a}\right )\right ) {\mathrm e}^{\frac {\int _{}^{x}\left (\operatorname {c1} \cosh \left (\lambda \operatorname {1} \textit {\_a} \right )^{\operatorname {n1}}+\operatorname {c2} \cosh \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \lambda \operatorname {2} }{a}\right )^{\operatorname {n2}}\right )d \textit {\_a}}{a}}\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.5.8.4 [1257] Problem 4
problem number 1257
Added April 3, 2019.
Problem Chapter 5.4.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a x \cosh (\lambda x+\mu y) w + b \cosh (\nu x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*Cosh[lambda*x+my*y]+b*Cosh[nu*x];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right )+\frac {a x \sinh (\lambda x+\text {my} y)}{\lambda x+\text {my} y}+b \text {Chi}(\nu x)\right \}\right \}\]
Maple ✓
restart;
pde := x*diff(w(x,y),x)+ y*diff(w(x,y),y) = a*x*cosh(lambda*x+my*y)+b*cosh(nu*x);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {a \sinh \left (\lambda x +\operatorname {my} y \right ) x}{\lambda x +\operatorname {my} y}+b \,\operatorname {Chi}\left (\nu x \right )+f_{1} \left (\frac {y}{x}\right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.5.8.5 [1258] Problem 5
problem number 1258
Added April 3, 2019.
Problem Chapter 5.4.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \cosh ^n(\lambda x) w_x + b \cosh ^m(\mu x) w_y = c \cosh ^k(\nu x) w + p \cosh ^s(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*Cosh[lambda*x]^n*D[w[x, y], x] + b*Cosh[mu*x]^m*D[w[x, y], y] == c*Cosh[nu*x]^k+p*Cosh[beta*y]^s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {\cosh ^{-n}(\lambda K[2]) \left (c \cosh ^k(\nu K[2])+p \cosh ^s\left (\beta \left (y-\int _1^x\frac {b \cosh ^{-n}(\lambda K[1]) \cosh ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \cosh ^{-n}(\lambda K[1]) \cosh ^m(\mu K[1])}{a}dK[1]\right )\right )\right )}{a}dK[2]+c_1\left (y-\int _1^x\frac {b \cosh ^{-n}(\lambda K[1]) \cosh ^m(\mu K[1])}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*cosh(lambda*x)^n*diff(w(x,y),x)+ b*cosh(mu*x)^m*diff(w(x,y),y) = c*cosh(nu*x)^k+p*cosh(beta*y)^s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = f_{1} \left (-\frac {b \int \cosh \left (\lambda x \right )^{-n} \cosh \left (\mu x \right )^{m}d x}{a}+y \right )+\frac {\int _{}^{x}\cosh \left (\lambda \textit {\_b} \right )^{-n} \left (c \cosh \left (\nu \textit {\_b} \right )^{k}+{\cosh \left (\frac {\beta \left (b \int \cosh \left (\lambda \textit {\_b} \right )^{-n} \cosh \left (\mu \textit {\_b} \right )^{m}d \textit {\_b} -b \int \cosh \left (\lambda x \right )^{-n} \cosh \left (\mu x \right )^{m}d x +y a \right )}{a}\right )}^{s} p \right )d \textit {\_b}}{a}\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.5.8.6 [1259] Problem 6
problem number 1259
Added April 3, 2019.
Problem Chapter 5.4.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \cosh ^n(\lambda x) w_x + b \cosh ^m(\mu x) w_y = c \cosh ^k(\nu y) w + p \cosh ^s(\beta x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*Cosh[lambda*x]^n*D[w[x, y], x] + b*Cosh[mu*x]^m*D[w[x, y], y] == c*Cosh[nu*y]^k+p*Cosh[beta*x]^s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {\cosh ^{-n}(\lambda K[2]) \left (c \cosh ^k\left (\nu \left (y-\int _1^x\frac {b \cosh ^{-n}(\lambda K[1]) \cosh ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \cosh ^{-n}(\lambda K[1]) \cosh ^m(\mu K[1])}{a}dK[1]\right )\right )+p \cosh ^s(\beta K[2])\right )}{a}dK[2]+c_1\left (y-\int _1^x\frac {b \cosh ^{-n}(\lambda K[1]) \cosh ^m(\mu K[1])}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*cosh(lambda*x)^n*diff(w(x,y),x)+ b*cosh(mu*x)^m*diff(w(x,y),y) = c*cosh(nu*y)^k+p*cosh(beta*x)^s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = f_{1} \left (-\frac {b \int \cosh \left (\mu x \right )^{m} \cosh \left (\lambda x \right )^{-n}d x}{a}+y \right )+\frac {\int _{}^{x}\cosh \left (\lambda \textit {\_b} \right )^{-n} \left ({\cosh \left (\frac {\nu \left (b \int \cosh \left (\mu \textit {\_b} \right )^{m} \cosh \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b} -b \int \cosh \left (\mu x \right )^{m} \cosh \left (\lambda x \right )^{-n}d x +y a \right )}{a}\right )}^{k} c +p \cosh \left (\beta \textit {\_b} \right )^{s}\right )d \textit {\_b}}{a}\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________