6.4.7 3.2

6.4.7.1 [1069] Problem 1

problem number 1069

Added Feb. 23, 2019.

Problem Chapter 4.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = (c y e^{\lambda x} + k x e^{\mu y} ) w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {-\left (-b \mu x +a \right ) a^{2} k \lambda ^{2} {\mathrm e}^{\mu y}+\left (a \lambda y -b \right ) b^{2} c \mu ^{2} {\mathrm e}^{\lambda x}}{a^{2} b^{2} \lambda ^{2} \mu ^{2}}}\]

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6.4.7.2 [1070] Problem 2

problem number 1070

Added Feb. 23, 2019.

Problem Chapter 4.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + y w_y = a x e^{\lambda x+\mu y} w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right ) e^{\frac {a x e^{\lambda x+\mu y}}{\lambda x+\mu y}}\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {y}{x}\right ) {\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x +\mu y}}{\lambda +\frac {\mu y}{x}}}\]

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6.4.7.3 [1071] Problem 3

problem number 1071

Added Feb. 23, 2019.

Problem Chapter 4.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + y w_y = (a y e^{\lambda x}+ b x e^{\mu y}) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right ) e^{\frac {a y e^{\lambda x}}{\lambda x}+\frac {b x e^{\mu y}}{\mu y}}\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {y}{x}\right ) {\mathrm e}^{\frac {\left (\frac {a \mu y^{2} {\mathrm e}^{\lambda x}}{x^{2}}+b \lambda \,{\mathrm e}^{\mu y}\right ) x}{\lambda \mu y}}\]

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6.4.7.4 [1072] Problem 4

problem number 1072

Added Feb. 23, 2019.

Problem Chapter 4.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x^k w_x + b e^{\lambda y} w_y = (c x^n+s) w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {b \lambda x^{-k +1}-\left (k -1\right ) a \,{\mathrm e}^{-\lambda y}}{\left (k -1\right ) b \lambda }\right ) {\mathrm e}^{-\frac {\left (\left (k -1\right ) c x^{n}+\left (k -n -1\right ) s \right ) x^{-k +1}}{\left (k -n -1\right ) \left (k -1\right ) a}}\]

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6.4.7.5 [1073] Problem 5

problem number 1073

Added Feb. 23, 2019.

Problem Chapter 4.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a y^k w_x + b e^{\lambda x} w_y = (c e^{\mu x}+s) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {y^{k+1}}{k+1}-\frac {b e^{\lambda x}}{a \lambda }\right ) \exp \left (-\frac {y^{k+1} \left (\left (y^{k+1}\right )^{\frac {1}{k+1}}\right )^{-k} \left (c \lambda e^{\mu x} \, _2F_1\left (1,\frac {\lambda +k \mu +\mu }{k \lambda +\lambda };\frac {\lambda +\mu }{\lambda };\frac {b e^{\lambda x} (k+1)}{b e^{\lambda x} (k+1)-a \lambda y^{k+1}}\right )-(k+1) \mu s \, _2F_1\left (1,\frac {1}{k+1};\frac {k+2}{k+1};\frac {a \lambda y^{k+1}}{a \lambda y^{k+1}-b e^{\lambda x} (k+1)}\right )\right )}{\mu \left (b (k+1) e^{\lambda x}-a \lambda y^{k+1}\right )}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a \lambda y y^{k}-\left (k +1\right ) b \,{\mathrm e}^{\lambda x}}{a \lambda }\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (c \,{\mathrm e}^{\mathit {\_a} \mu }+s \right ) \left (\left (\frac {a \lambda y^{k +1}+\left (k +1\right ) b \,{\mathrm e}^{\mathit {\_a} \lambda }-\left (k +1\right ) b \,{\mathrm e}^{\lambda x}}{a \lambda }\right )^{\frac {1}{k +1}}\right )^{-k}}{a}d\mathit {\_a}}\]

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6.4.7.6 [1074] Problem 6

problem number 1074

Added Feb. 23, 2019.

Problem Chapter 4.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a e^{\lambda x} w_x + b y^k w_y = (c x^n+s) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {b e^{-\lambda x}}{a \lambda }-\frac {y^{1-k}}{k-1}\right ) \exp \left (-\frac {c x^n (\lambda x)^{-n} \operatorname {Gamma}(n+1,\lambda x)+s e^{-\lambda x}}{a \lambda }\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a \lambda y^{-k +1}-\left (k -1\right ) b \,{\mathrm e}^{-\lambda x}}{a \lambda }\right ) {\mathrm e}^{\frac {c x^{n} \left (\lambda x \right )^{-\frac {n}{2}} \WhittakerM \left (\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \lambda x \right ) {\mathrm e}^{-\frac {\lambda x}{2}}-\left ({\mathrm e}^{-\lambda x}-1\right ) \left (n +1\right ) s}{\left (n +1\right ) a \lambda }}\]

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6.4.7.7 [1075] Problem 7

problem number 1075

Added Feb. 23, 2019.

Problem Chapter 4.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a e^{\lambda y} w_x + b x^k w_y = (c e^{\mu x}+s) w \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-b \lambda x^{k +1}+\left (k +1\right ) a \,{\mathrm e}^{\lambda y}}{\left (k +1\right ) b \lambda }\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (c \,{\mathrm e}^{\mathit {\_a} \mu }+s \right ) \left (k +1\right )}{b \lambda \mathit {\_a}^{k +1}-b \lambda x^{k +1}+\left (k +1\right ) a \,{\mathrm e}^{\lambda y}}d\mathit {\_a}}\]

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