118 HFOPDE, chapter 4.8.1

118.1 Problem 1
118.2 Problem 2
118.3 Problem 3
118.4 Problem 4
118.5 Problem 5
118.6 Problem 6
118.7 Problem 7
118.8 Problem 8
118.9 Problem 9
118.10 Problem 10
118.11 Problem 11
118.12 Problem 12
118.13 Problem 13
118.14 Problem 14
118.15 Problem 15

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118.1 Problem 1

problem number 953

Added March 10, 2019.

Problem Chapter 4.8.1.1, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a w_x + b w_y = f(x) w$

Mathematica

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

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ){{\rm e}^{\int \!{\frac{f \left ( x \right ) }{a}}\,{\rm d}x}}$

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118.2 Problem 2

problem number 954

Added March 10, 2019.

Problem Chapter 4.8.1.2, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + a w_y = f(x) y w$

Mathematica

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

Maple

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

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118.3 Problem 3

problem number 955

Added March 10, 2019.

Problem Chapter 4.8.1.3, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + a w_y = (f(x) y^2+g(x) y+h(x)) w$

Mathematica

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

Maple

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

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118.4 Problem 4

problem number 956

Added March 10, 2019.

Problem Chapter 4.8.1.4, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + a w_y = f(x) y^k w$

Mathematica

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

Maple

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

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118.5 Problem 5

problem number 957

Added March 10, 2019.

Problem Chapter 4.8.1.5, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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118.6 Problem 6

problem number 958

Added March 10, 2019.

Problem Chapter 4.8.1.6, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + (a y + f(x)) w_y = g(x) w$

Mathematica

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

Maple

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

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118.7 Problem 7

problem number 959

Added March 10, 2019.

Problem Chapter 4.8.1.7, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + (a y + f(x)) w_y = g(x) y^k w$

Mathematica

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

Maple

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

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118.8 Problem 8

problem number 960

Added March 10, 2019.

Problem Chapter 4.8.1.8, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$f(x) w_x + y^k w_y = g(x) w$

Mathematica

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

Maple

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

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118.9 Problem 9

problem number 961

Added March 10, 2019.

Problem Chapter 4.8.1.9, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$f(x) w_x + (y+a) w_y = (b y+c) w$

Mathematica

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

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( \left ( y+a \right ){{\rm e}^{-\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x}} \right ){{\rm e}^{c\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+b \left ( y+a \right ) -ab\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x}}$

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118.10 Problem 10

problem number 962

Added March 10, 2019.

Problem Chapter 4.8.1.10, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$f(x) w_x + (y+a x) w_y = g(x) w$

Mathematica

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

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -a\int \!{\frac{x{{\rm e}^{-\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x}}}{f \left ( x \right ) }}\,{\rm d}x+y{{\rm e}^{-\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x}} \right ){{\rm e}^{\int \!{\frac{g \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}$

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118.11 Problem 11

problem number 963

Added March 10, 2019.

Problem Chapter 4.8.1.11, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$f(x) w_x + (g_1(x) y+g_0(x)) w_y = \left ( h_2(x) y^2+ h_1(x)y + h_0(x) \right ) w$

Mathematica

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

Maple

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

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118.12 Problem 12

problem number 964

Added March 10, 2019.

Problem Chapter 4.8.1.12, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$f(x) w_x + (g_1(x) y+g_2(x) y^k) w_y = h(x) w$

Mathematica

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

Maple

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

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118.13 Problem 13

problem number 965

Added March 10, 2019.

Problem Chapter 4.8.1.13, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

$\text{DSolve}\left [f(x) w^{(1,0)}(x,y)+w^{(0,1)}(x,y) \left (y \text{g1}(x)+\text{g2}(x) e^{\lambda y}\right )=h(x) w(x,y),w(x,y),\{x,y\}\right ]$

Maple

$\text{ sol=() }$

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118.14 Problem 14

problem number 966

Added March 10, 2019.

Problem Chapter 4.8.1.14, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$f(x) y^k w_x + g(x) w_y = h(x) w$

Mathematica

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

Maple

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

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118.15 Problem 15

problem number 967

Added March 10, 2019.

Problem Chapter 4.8.1.15, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$f(x) e^{\lambda y} w_x + g(x) w_y = h(x) w$

Mathematica

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

Maple

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