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'));
 

\[w \left ( x,y \right ) ={\frac {1}{{\mu }^{2}{b}^{2}{a}^{2}{\lambda }^{2}} \left ( {\it \_F1} \left ( {\frac {ay-bx}{a}} \right ) {\mu }^{2}{b}^{2}{a}^{2}{\lambda }^{2}+{b}^{2}c{\mu }^{2} \left ( y\lambda \,a-b \right ) {{\rm e}^{\lambda \,x}}-{a}^{2}{{\rm e}^{\mu \,y}}k{\lambda }^{2} \left ( -b\mu \,x+a \right ) \right ) }\]

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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]];
 

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

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'));
 

\[w \left ( x,y \right ) ={\frac { \left ( -a\lambda \,x \right ) ^{-k}k{x}^{k} \left ( \Gamma \left ( k,-a\lambda \,x \right ) -\Gamma \left ( k \right ) \right ) {{\rm e}^{\lambda \, \left ( -ax+y \right ) }}+{x}^{k}{{\rm e}^{y\lambda }}+{\it \_F1} \left ( -ax+y \right ) \lambda }{\lambda }}\]

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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'));
 

\[w \left ( x,y \right ) ={\frac {1}{\beta } \left ( c{{\rm e}^{\beta \,x}}+{\it \_F1} \left ( {\frac { \left ( y \left ( a-\lambda \right ) {{\rm e}^{x \left ( a-\lambda \right ) }}+{{\rm e}^{ax}}b \right ) {{\rm e}^{-x \left ( 2\,a-\lambda \right ) }}}{a-\lambda }} \right ) \beta \right ) }\]

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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'));
 

\[w \left ( x,y \right ) ={\frac {1}{\mu } \left ( {\it \_F1} \left ( {\frac {1}{{y}^{k}} \left ( {{\rm e}^{{\frac {a{{\rm e}^{\lambda \,x}}}{\lambda }}}}b{y}^{k} \left ( k-1 \right ) \int \!{{\rm e}^{{\frac {a{{\rm e}^{\lambda \,x}} \left ( k-1 \right ) +\beta \,x\lambda }{\lambda }}}}\,{\rm d}x+{{\rm e}^{{\frac {a{{\rm e}^{\lambda \,x}}k}{\lambda }}}}y \right ) \left ( {{\rm e}^{{\frac {a{{\rm e}^{\lambda \,x}}}{\lambda }}}} \right ) ^{-1}} \right ) \mu +c{{\rm e}^{\mu \,x}} \right ) }\]

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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]];
 

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

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'));
 

\[w \left ( x,y \right ) ={\frac {1}{\beta } \left ( c{{\rm e}^{\beta \,x}}+{\it \_F1} \left ( {\frac {1}{a\lambda \, \left ( n+1 \right ) \left ( k+n+2 \right ) \left ( 3+2\,k+n \right ) } \left ( -{{\rm e}^{{\frac {{x}^{k+1}\lambda \,a}{2\,k+2}}}} \left ( k+1 \right ) ^{2} \left ( \left ( -k-n-2 \right ) {x}^{n-k}+a{x}^{n+1}\lambda \right ) \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{{\frac {-k-n-2}{2\,k+2}}}b \WhittakerM \left ( {\frac {n-k}{2\,k+2}},{\frac {3+2\,k+n}{2\,k+2}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) -2\, \left ( -1/2\,{x}^{n-k} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{{\frac {-k-n-2}{2\,k+2}}}{{\rm e}^{{\frac {{x}^{k+1}\lambda \,a}{2\,k+2}}}}b \left ( k+1 \right ) \left ( k+n+2 \right ) \WhittakerM \left ( {\frac {k+n+2}{2\,k+2}},{\frac {3+2\,k+n}{2\,k+2}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) + \left ( k+n/2+3/2 \right ) a \left ( n+1 \right ) {{\rm e}^{{\frac { \left ( {x}^{k+1}a-y \left ( k+1 \right ) \right ) \lambda }{k+1}}}} \right ) \left ( k+n+2 \right ) \right ) } \right ) \beta \right ) }\]

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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'));
 

\[w \left ( x,y \right ) ={a{{\rm e}^{\lambda \,x+\mu \,y}} \left ( {\frac {\mu \,y}{x}}+\lambda \right ) ^{-1}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \]

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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'));
 

\[w \left ( x,y \right ) ={\frac {bx{{\rm e}^{\mu \,y}}}{\mu \,y}}+{\frac {ay{{\rm e}^{\lambda \,x}}}{\lambda \,x}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \]

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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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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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'));
 

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

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