6.3.7 3.2

6.3.7.1 [888] Problem 1

problem number 888

Added Feb. 9, 2019.

Problem Chapter 3.3.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c y e^{\lambda x} + k x e^{\mu y} \]

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

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

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); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.3.7.2 [889] Problem 2

problem number 889

Added Feb. 9, 2019.

Problem Chapter 3.3.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = a x^k e^{\lambda y} \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \frac {f_{1} \left (-a x +y \right ) \lambda +\left ({\mathrm e}^{\lambda y}+\left (-a \lambda x \right )^{-k} {\mathrm e}^{-\left (a x -y \right ) \lambda } \left (\Gamma \left (k , -a \lambda x \right ) k -\Gamma \left (k +1\right )\right )\right ) x^{k}}{\lambda }\]

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6.3.7.3 [890] Problem 3

problem number 890

Added Feb. 9, 2019.

Problem Chapter 3.3.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y+b e^{\lambda x}) w_y = c e^{\beta x} \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.7.4 [891] Problem 4

problem number 891

Added Feb. 9, 2019.

Problem Chapter 3.3.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.7.5 [892] Problem 5

problem number 892

Added Feb. 9, 2019.

Problem Chapter 3.3.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a x^k+b x^n e^{\lambda y}) w_y = c e^{\beta x} \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \frac {c \,{\mathrm e}^{\beta x}+f_{1} \left (-\frac {x^{-k} \left ({\mathrm e}^{\frac {x a \,x^{k} \lambda }{2 k +2}} b \,x^{n} \left (k +1\right )^{2} \left (x a \,x^{k} \lambda -k -n -2\right ) \operatorname {WhittakerM}\left (\frac {-k +n}{2 k +2}, \frac {2 k +n +3}{2 k +2}, -\frac {x a \,x^{k} \lambda }{k +1}\right )+2 \left (-\frac {\left (k +1\right ) \left (k +n +2\right ) {\mathrm e}^{\frac {x a \,x^{k} \lambda }{2 k +2}} b \,x^{n} \operatorname {WhittakerM}\left (\frac {k +n +2}{2 k +2}, \frac {2 k +n +3}{2 k +2}, -\frac {x a \,x^{k} \lambda }{k +1}\right )}{2}+x^{k} {\mathrm e}^{\frac {\lambda \left (a x \,x^{k}-y \left (k +1\right )\right )}{k +1}} \left (-\frac {x a \,x^{k} \lambda }{k +1}\right )^{\frac {k +n +2}{2 k +2}} \left (k +\frac {n}{2}+\frac {3}{2}\right ) a \left (n +1\right )\right ) \left (k +n +2\right )\right ) \left (-\frac {x a \,x^{k} \lambda }{k +1}\right )^{\frac {-k -n -2}{2 k +2}}}{a \lambda \left (n +1\right ) \left (k +n +2\right ) \left (2 k +n +3\right )}\right ) \beta }{\beta }\]

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6.3.7.6 [893] Problem 6

problem number 893

Added Feb. 9, 2019.

Problem Chapter 3.3.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a x e^{\lambda x+ \mu y} \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \frac {a \,{\mathrm e}^{\lambda x +\mu y} x}{\lambda x +\mu y}+f_{1} \left (\frac {y}{x}\right )\]

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6.3.7.7 [894] Problem 7

problem number 894

Added Feb. 9, 2019.

Problem Chapter 3.3.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a y e^{\lambda x} + b x e^{\mu y} \]

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

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

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); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.3.7.8 [895] Problem 8

problem number 895

Added Feb. 9, 2019.

Problem Chapter 3.3.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^k w_x + b e^{\lambda y} w_y = c x^n+s \]

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

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

Maple ✓

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

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

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6.3.7.9 [896] Problem 9

problem number 896

Added Feb. 9, 2019.

Problem Chapter 3.3.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^k w_x + b e^{\lambda x} w_y = c e^{\mu x}+s \]

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

\[\left \{\left \{w(x,y)\to \frac {y^{k+1} \left (\left (y^{k+1}\right )^{\frac {1}{k+1}}\right )^{-k} \left ((k+1) \mu s \operatorname {Hypergeometric2F1}\left (1,\frac {1}{k+1},\frac {k+2}{k+1},\frac {a \lambda y^{k+1}}{a \lambda y^{k+1}-b e^{\lambda x} (k+1)}\right )-c \lambda e^{\mu x} \operatorname {Hypergeometric2F1}\left (1,\frac {\lambda +k \mu +\mu }{k \lambda +\lambda },\frac {\lambda +\mu }{\lambda },\frac {b e^{\lambda x} (k+1)}{b e^{\lambda x} (k+1)-a \lambda y^{k+1}}\right )\right )}{\mu \left (b (k+1) e^{\lambda x}-a \lambda y^{k+1}\right )}+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b e^{\lambda x}}{a \lambda }\right )\right \}\right \}\]

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; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.3.7.10 [897] Problem 10

problem number 897

Added Feb. 9, 2019.

Problem Chapter 3.3.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda x} w_x + b y^k w_y = c x^n+s \]

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

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

Maple ✓

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

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

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6.3.7.11 [898] Problem 11

problem number 898

Added Feb. 9, 2019.

Problem Chapter 3.3.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda y} w_x + b x^k w_y = c Exp[\mu x]+s \]

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

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

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; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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