91 HFOPDE, chapter 3.8.1

 91.1 Problem 1
 91.2 Problem 2
 91.3 Problem 3
 91.4 Problem 4
 91.5 Problem 5
 91.6 Problem 6
 91.7 Problem 7
 91.8 Problem 8
 91.9 Problem 9
 91.10 Problem 10
 91.11 Problem 11
 91.12 Problem 12
 91.13 Problem 13
 91.14 Problem 14
 91.15 Problem 15
 91.16 Problem 16

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91.1 Problem 1

problem number 775

Added Feb. 11, 2019.

Problem Chapter 3.8.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = f(x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to \int _1^x \frac{f(K[1])}{a} \, dK[1]+c_1\left (\frac{a y-b x}{a}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int \!{\frac{f \left ( x \right ) }{a}}\,{\rm d}x+{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \]

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91.2 Problem 2

problem number 776

Added Feb. 11, 2019.

Problem Chapter 3.8.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = y f(x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to \int _1^x f(K[1]) (a K[1]-a x+y) \, dK[1]+c_1(y-a x)\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!f \left ({\it \_a} \right ) \left ({\it \_a}\,a-ax+y \right ){d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \]

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91.3 Problem 3

problem number 777

Added Feb. 11, 2019.

Problem Chapter 3.8.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = y^2 f(x)+ y g(x) + h(x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to \int _1^x \left (f(K[1]) (a K[1]-a x+y)^2+g(K[1]) (a K[1]-a x+y)+h(K[1])\right ) \, dK[1]+c_1(y-a x)\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int ^{x}\! \left ( \left ( x-{\it \_a} \right ) a-y \right ) ^{2}f \left ({\it \_a} \right ) + \left ( \left ( -x+{\it \_a} \right ) a+y \right ) g \left ({\it \_a} \right ) +h \left ({\it \_a} \right ){d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \]

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91.4 Problem 4

problem number 778

Added Feb. 11, 2019.

Problem Chapter 3.8.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = y^k f(x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to \int _1^x f(K[1]) (a K[1]-a x+y)^k \, dK[1]+c_1(y-a x)\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int ^{x}\! \left ({\it \_a}\,a-ax+y \right ) ^{k}f \left ({\it \_a} \right ){d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \]

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91.5 Problem 5

problem number 779

Added Feb. 11, 2019.

Problem Chapter 3.8.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to \int _1^x f(K[1]) e^{\lambda (a K[1]-a x+y)} \, dK[1]+c_1(y-a x)\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!f \left ({\it \_a} \right ){{\rm e}^{\lambda \,a{\it \_a}+\lambda \, \left ( -ax+y \right ) }}{d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \]

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91.6 Problem 6

problem number 780

Added Feb. 11, 2019.

Problem Chapter 3.8.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y + f(x) ) w_y = g(x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-e^{-a x} \left (e^{a x} \int _1^x e^{-a K[1]} f(K[1]) \, dK[1]-y\right )\right )+\int _1^x g(K[2]) \, dK[2]\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int \!g \left ( x \right ) \,{\rm d}x+{\it \_F1} \left ( -\int \!f \left ( x \right ){{\rm e}^{-ax}}\,{\rm d}x+y{{\rm e}^{-ax}} \right ) \]

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91.7 Problem 7

problem number 781

Added Feb. 11, 2019.

Problem Chapter 3.8.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y + f(x) ) w_y = y^k g(x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to \int _1^x g(K[2]) \left (e^{a K[2]} \left (\text{Integrate}\left [e^{-a K[1]} f(K[1]),\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]-e^{-a x} \left (e^{a x} \text{Integrate}\left [e^{-a K[1]} f(K[1]),\{K[1],1,x\},\text{Assumptions}\to \text{True}\right ]-y\right )\right )\right )^k \, dK[2]+c_1\left (-e^{-a x} \left (e^{a x} \int _1^x e^{-a K[1]} f(K[1]) \, dK[1]-y\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int ^{x}\! \left ( \left ( \int \!f \left ({\it \_b} \right ){{\rm e}^{-a{\it \_b}}}\,{\rm d}{\it \_b}-\int \!f \left ( x \right ){{\rm e}^{-ax}}\,{\rm d}x+y{{\rm e}^{-ax}} \right ){{\rm e}^{a{\it \_b}}} \right ) ^{k}g \left ({\it \_b} \right ){d{\it \_b}}+{\it \_F1} \left ( -\int \!f \left ( x \right ){{\rm e}^{-ax}}\,{\rm d}x+y{{\rm e}^{-ax}} \right ) \]

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91.8 Problem 8

problem number 782

Added Feb. 11, 2019.

Problem Chapter 3.8.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + y^k w_y = g(x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac{y^{-k} \left (k y^k \left (\int _1^x \frac{1}{f(K[1])} \, dK[1]\right )-y^k \left (\int _1^x \frac{1}{f(K[1])} \, dK[1]\right )+y\right )}{k-1}\right )+\int _1^x \frac{g(K[2])}{f(K[2])} \, dK[2]\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int \!g \left ( x \right ) \,{\rm d}x+{\it \_F1} \left ({\frac{y}{{y}^{k}}}+kx-x \right ) \]

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91.9 Problem 9

problem number 783

Added Feb. 11, 2019.

Problem Chapter 3.8.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (y+a) w_y = b y+c \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) = \left ( -ab+c \right ) x+ab+by+{\it \_F1} \left ( \left ( a+y \right ){{\rm e}^{-x}} \right ) \]

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91.10 Problem 10

problem number 784

Added Feb. 11, 2019.

Problem Chapter 3.8.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (y+a x) w_y = g(x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-e^{-\int _1^x \frac{1}{f(K[1])} \, dK[1]} \left (e^{\int _1^x \frac{1}{f(K[1])} \, dK[1]} \int _1^x \frac{a K[2] \exp \left (-\text{Integrate}\left [\frac{1}{f(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{f(K[2])} \, dK[2]-y\right )\right )+\int _1^x \frac{g(K[3])}{f(K[3])} \, dK[3]\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int \!g \left ( x \right ) \,{\rm d}x+{\it \_F1} \left ( \left ( ax+a+y \right ){{\rm e}^{-x}} \right ) \]

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91.11 Problem 11

problem number 785

Added Feb. 11, 2019.

Problem Chapter 3.8.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (y g_1(x)+g_0(x)) w_y = y^2 h_2(x)+y h_1(x) + h_0(x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (y e^{-\int _1^x \frac{\text{g1}(K[1])}{f(K[1])} \, dK[1]}-\int _1^x \frac{\text{g0}(K[2]) \exp \left (-\text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{f(K[2])} \, dK[2]\right )+\int _1^x \frac{\text{h1}(K[3]) \exp \left (\text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[3]\},\text{Assumptions}\to \text{True}\right ]\right ) \left (-\text{Integrate}\left [\frac{\text{g0}(K[2]) \exp \left (-\text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{f(K[2])},\{K[2],1,x\},\text{Assumptions}\to \text{True}\right ]+\text{Integrate}\left [\frac{\text{g0}(K[2]) \exp \left (-\text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{f(K[2])},\{K[2],1,K[3]\},\text{Assumptions}\to \text{True}\right ]+y \exp \left (-\text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text{Assumptions}\to \text{True}\right ]\right )\right )+\text{h2}(K[3]) \exp \left (2 \text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[3]\},\text{Assumptions}\to \text{True}\right ]\right ) \left (-\text{Integrate}\left [\frac{\text{g0}(K[2]) \exp \left (-\text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{f(K[2])},\{K[2],1,x\},\text{Assumptions}\to \text{True}\right ]+\text{Integrate}\left [\frac{\text{g0}(K[2]) \exp \left (-\text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{f(K[2])},\{K[2],1,K[3]\},\text{Assumptions}\to \text{True}\right ]+y \exp \left (-\text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text{Assumptions}\to \text{True}\right ]\right )\right )^2+\text{h0}(K[3])}{f(K[3])} \, dK[3]\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\it h2} \left ({\it \_f} \right ) \left ( \int \!{\it g0} \left ({\it \_f} \right ){{\rm e}^{-\int \!{\it g1} \left ({\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f} \right ) ^{2}{{\rm e}^{2\,\int \!{\it g1} \left ({\it \_f} \right ) \,{\rm d}{\it \_f}}}+2\,{\it h2} \left ({\it \_f} \right ) \int \!{\it g0} \left ({\it \_f} \right ){{\rm e}^{-\int \!{\it g1} \left ({\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}{{\rm e}^{2\,\int \!{\it g1} \left ({\it \_f} \right ) \,{\rm d}{\it \_f}}} \left ( -\int \!{\it g0} \left ( x \right ){{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) +{\it h2} \left ({\it \_f} \right ){{\rm e}^{2\,\int \!{\it g1} \left ({\it \_f} \right ) \,{\rm d}{\it \_f}}} \left ( -\int \!{\it g0} \left ( x \right ){{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) ^{2}+{{\rm e}^{\int \!{\it g1} \left ({\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it h1} \left ({\it \_f} \right ) \int \!{\it g0} \left ({\it \_f} \right ){{\rm e}^{-\int \!{\it g1} \left ({\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}+{{\rm e}^{\int \!{\it g1} \left ({\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it h1} \left ({\it \_f} \right ) \left ( -\int \!{\it g0} \left ( x \right ){{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) +{\it h0} \left ({\it \_f} \right ){d{\it \_f}}+{\it \_F1} \left ( -\int \!{\it g0} \left ( x \right ){{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) \]

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91.12 Problem 12

problem number 786

Added Feb. 11, 2019.

Problem Chapter 3.8.1.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (y g_1(x)+y^k g_2(x)) w_y = h(x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left ((k-1) \int _1^x \frac{\text{g2}(K[2]) \exp \left ((k-1) \text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{f(K[2])} \, dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x \frac{\text{g1}(K[1])}{f(K[1])} \, dK[1]\right )\right )+\int _1^x \frac{h(K[3])}{f(K[3])} \, dK[3]\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int \!h \left ( x \right ) \,{\rm d}x+{\it \_F1} \left ({y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it g1} \left ( x \right ) \,{\rm d}x}}+k\int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it g1} \left ( x \right ) \,{\rm d}x}}{\it g2} \left ( x \right ) \,{\rm d}x-\int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it g1} \left ( x \right ) \,{\rm d}x}}{\it g2} \left ( x \right ) \,{\rm d}x \right ) \]

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91.13 Problem 13

problem number 787

Added Feb. 11, 2019.

Problem Chapter 3.8.1.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (g_1(x)+e^{\lambda y} g_2(x)) w_y = h(x) \]

Mathematica

\[ \text{DSolve}\left [f(x) w^{(1,0)}(x,y)+w^{(0,1)}(x,y) \left (\text{g1}(x)+e^{\lambda y}\right )=h(x),w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) =\int \!{\frac{h \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x+{\it \_F1} \left ( -{\frac{1}{\lambda } \left ( \lambda \,\int \!{\frac{1}{f \left ( x \right ) }{{\rm e}^{\lambda \,\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+{{\rm e}^{-\lambda \, \left ( y-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \right ) }} \right ) } \right ) \]

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91.14 Problem 14

problem number 788

Added Feb. 11, 2019.

Problem Chapter 3.8.1.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ y^k f(x) w_x + g(x) w_y = h(x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to \int _1^x \frac{h(K[2]) \left (\left (-(k+1) \text{Integrate}\left [\frac{g(K[1])}{f(K[1])},\{K[1],1,x\},\text{Assumptions}\to \text{True}\right ]+(k+1) \text{Integrate}\left [\frac{g(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]+y^{k+1}\right )^{\frac{1}{k+1}}\right )^{-k}}{f(K[2])} \, dK[2]+c_1\left (\frac{y^{k+1}}{k+1}-\int _1^x \frac{g(K[1])}{f(K[1])} \, dK[1]\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{h \left ({\it \_b} \right ) }{f \left ({\it \_b} \right ) } \left ( \left ( k\int \!{\frac{g \left ({\it \_b} \right ) }{f \left ({\it \_b} \right ) }}\,{\rm d}{\it \_b}+{y}^{k}y-k\int \!{\frac{g \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x-\int \!{\frac{g \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x+\int \!{\frac{g \left ({\it \_b} \right ) }{f \left ({\it \_b} \right ) }}\,{\rm d}{\it \_b} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}}{d{\it \_b}}+{\it \_F1} \left ({y}^{k}y-k\int \!{\frac{g \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x-\int \!{\frac{g \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \right ) \]

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91.15 Problem 15

problem number 789

Added Feb. 11, 2019.

Problem Chapter 3.8.1.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ y^k f(x) w_x + (y^{k+1} g_1(x) + g_0(x)) w_y = y^{3 k +1} h_2(x) + y^{2 k+1} h_1(x) + y^k h_0(x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (y^{k+1} \exp \left (-(k+1) \int _1^x \frac{\text{g1}(K[1])}{f(K[1])} \, dK[1]\right )-(k+1) \int _1^x \frac{\text{g0}(K[2]) \exp \left (-(k+1) \text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{f(K[2])} \, dK[2]\right )+\int _1^x \frac{\text{h1}(K[3]) \left (\left (\exp \left (-(k+1) \left (\text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text{Assumptions}\to \text{True}\right ]-\text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[3]\},\text{Assumptions}\to \text{True}\right ]\right )\right ) \left (-(k+1) \exp \left ((k+1) \text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text{Assumptions}\to \text{True}\right ]\right ) \text{Integrate}\left [\frac{\text{g0}(K[2]) \exp \left (-(k+1) \text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{f(K[2])},\{K[2],1,x\},\text{Assumptions}\to \text{True}\right ]+(k+1) \text{Integrate}\left [\frac{\text{g0}(K[2]) \exp \left (-(k+1) \text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{f(K[2])},\{K[2],1,K[3]\},\text{Assumptions}\to \text{True}\right ] \exp \left ((k+1) \text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text{Assumptions}\to \text{True}\right ]\right )+y^{k+1}\right )\right )^{\frac{1}{k+1}}\right )^{k+1}+\text{h2}(K[3]) \left (\left (\exp \left (-(k+1) \left (\text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text{Assumptions}\to \text{True}\right ]-\text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[3]\},\text{Assumptions}\to \text{True}\right ]\right )\right ) \left (-(k+1) \exp \left ((k+1) \text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text{Assumptions}\to \text{True}\right ]\right ) \text{Integrate}\left [\frac{\text{g0}(K[2]) \exp \left (-(k+1) \text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{f(K[2])},\{K[2],1,x\},\text{Assumptions}\to \text{True}\right ]+(k+1) \text{Integrate}\left [\frac{\text{g0}(K[2]) \exp \left (-(k+1) \text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{f(K[2])},\{K[2],1,K[3]\},\text{Assumptions}\to \text{True}\right ] \exp \left ((k+1) \text{Integrate}\left [\frac{\text{g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text{Assumptions}\to \text{True}\right ]\right )+y^{k+1}\right )\right )^{\frac{1}{k+1}}\right )^{2 k+1}+\text{h0}(K[3])}{f(K[3])} \, dK[3]\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{f \left ({\it \_f} \right ) } \left ( \left ( \left ( k\int \!{\frac{{\it g0} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f}+{y}^{k+1}{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}-k\int \!{\frac{{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}}\,{\rm d}x-\int \!{\frac{{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}}\,{\rm d}x+\int \!{\frac{{\it g0} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f} \right ) ^{ \left ( k+1 \right ) ^{-1}}{{\rm e}^{\int \!{\frac{{\it g1} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f}}} \right ) ^{2\,k} \left ( k\int \!{\frac{{\it g0} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f}+{y}^{k+1}{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}-k\int \!{\frac{{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}}\,{\rm d}x-\int \!{\frac{{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}}\,{\rm d}x+\int \!{\frac{{\it g0} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f} \right ) ^{ \left ( k+1 \right ) ^{-1}}{{\rm e}^{\int \!{\frac{{\it g1} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f}}}{\it h2} \left ({\it \_f} \right ) + \left ( \left ( k\int \!{\frac{{\it g0} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f}+{y}^{k+1}{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}-k\int \!{\frac{{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}}\,{\rm d}x-\int \!{\frac{{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}}\,{\rm d}x+\int \!{\frac{{\it g0} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f} \right ) ^{ \left ( k+1 \right ) ^{-1}}{{\rm e}^{\int \!{\frac{{\it g1} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f}}} \right ) ^{k} \left ( k\int \!{\frac{{\it g0} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f}+{y}^{k+1}{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}-k\int \!{\frac{{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}}\,{\rm d}x-\int \!{\frac{{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}}\,{\rm d}x+\int \!{\frac{{\it g0} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f} \right ) ^{ \left ( k+1 \right ) ^{-1}}{{\rm e}^{\int \!{\frac{{\it g1} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f}}}{\it h1} \left ({\it \_f} \right ) +{\it h0} \left ({\it \_f} \right ) \right ) }{d{\it \_f}}+{\it \_F1} \left ({y}^{k+1}{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}-k\int \!{\frac{{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}}\,{\rm d}x-\int \!{\frac{{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac{{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}}\,{\rm d}x \right ) \]

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91.16 Problem 16

problem number 790

Added Feb. 11, 2019.

Problem Chapter 3.8.1.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) e^{\lambda x} w_x + g(x) w_y = h(x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (y-\int _1^x \frac{g(K[1]) e^{-\lambda K[1]}}{f(K[1])} \, dK[1]\right )+\int _1^x \frac{h(K[2]) e^{-\lambda K[2]}}{f(K[2])} \, dK[2]\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int \!{\frac{h \left ( x \right ){{\rm e}^{-\lambda \,x}}}{f \left ( x \right ) }}\,{\rm d}x+{\it \_F1} \left ( -\int \!{\frac{g \left ( x \right ){{\rm e}^{-\lambda \,x}}}{f \left ( x \right ) }}\,{\rm d}x+y \right ) \]