6.4.24 8.1

   6.4.24.1 [1161] Problem 1
   6.4.24.2 [1162] Problem 2
   6.4.24.3 [1163] Problem 3
   6.4.24.4 [1164] Problem 4
   6.4.24.5 [1165] Problem 5
   6.4.24.6 [1166] Problem 6
   6.4.24.7 [1167] Problem 7
   6.4.24.8 [1168] Problem 8
   6.4.24.9 [1169] Problem 9
   6.4.24.10 [1170] Problem 10
   6.4.24.11 [1171] Problem 11
   6.4.24.12 [1172] Problem 12
   6.4.24.13 [1173] Problem 13
   6.4.24.14 [1174] Problem 14
   6.4.24.15 [1175] Problem 15

6.4.24.1 [1161] Problem 1

problem number 1161

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

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

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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6.4.24.2 [1162] Problem 2

problem number 1162

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

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

Maple

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

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6.4.24.3 [1163] Problem 3

problem number 1163

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

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

Maple

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

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6.4.24.4 [1164] Problem 4

problem number 1164

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

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

Maple

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

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6.4.24.5 [1165] Problem 5

problem number 1165

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

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

Maple

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

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6.4.24.6 [1166] Problem 6

problem number 1166

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

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

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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6.4.24.7 [1167] Problem 7

problem number 1167

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

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

Maple

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

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6.4.24.8 [1168] Problem 8

problem number 1168

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

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

Maple

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

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6.4.24.9 [1169] Problem 9

problem number 1169

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

\[\left \{\left \{w(x,y)\to c_1\left (y \exp \left (-\int _1^x\frac{1}{f(K[1])}dK[1]\right )-\int _1^x\frac{a \exp \left (-\int _1^{K[2]}\frac{1}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right ) \exp \left (\int _1^x\frac{c+b \exp \left (\int _1^{K[3]}\frac{1}{f(K[1])}dK[1]\right ) \left (\exp \left (-\int _1^x\frac{1}{f(K[1])}dK[1]\right ) y-\int _1^x\frac{a \exp \left (-\int _1^{K[2]}\frac{1}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]+\int _1^{K[3]}\frac{a \exp \left (-\int _1^{K[2]}\frac{1}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right )}{f(K[3])}dK[3]\right )\right \}\right \}\]

Maple

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

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6.4.24.10 [1170] Problem 10

problem number 1170

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

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

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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6.4.24.11 [1171] Problem 11

problem number 1171

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

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

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}+{{\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 ) \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}^{\int \!{\frac{{\it g1} \left ({\it \_f} \right ) }{f \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f}}}{\it h1} \left ({\it \_f} \right ) \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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6.4.24.12 [1172] Problem 12

problem number 1172

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

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

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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6.4.24.13 [1173] Problem 13

problem number 1173

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

Failed

Maple

sol=()

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6.4.24.14 [1174] Problem 14

problem number 1174

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

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

Maple

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

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6.4.24.15 [1175] Problem 15

problem number 1175

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

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

Maple

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

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