6.4.25 8.2

   6.4.25.1 [1176] Problem 1
   6.4.25.2 [1177] Problem 2
   6.4.25.3 [1178] Problem 3
   6.4.25.4 [1179] Problem 4
   6.4.25.5 [1180] Problem 5
   6.4.25.6 [1181] Problem 6
   6.4.25.7 [1182] Problem 7

6.4.25.1 [1176] Problem 1

problem number 1176

Added March 10, 2019.

Problem Chapter 4.8.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.25.2 [1177] Problem 2

problem number 1177

Added March 10, 2019.

Problem Chapter 4.8.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.25.3 [1178] Problem 3

problem number 1178

Added March 10, 2019.

Problem Chapter 4.8.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.25.4 [1179] Problem 4

problem number 1179

Added March 10, 2019.

Problem Chapter 4.8.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + g(y) w_y = (h_1(x)+h_2(y)) w \]

Mathematica

Failed

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y \right ){{\rm e}^{\int ^{x}\!{\frac{{\it h1} \left ({\it \_f} \right ) +{\it h2} \left ( \RootOf \left ( \int \! \left ( f \left ({\it \_f} \right ) \right ) ^{-1}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\! \left ( g \left ({\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}-\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y \right ) \right ) }{f \left ({\it \_f} \right ) }}{d{\it \_f}}}}\] contains RootOf

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6.4.25.5 [1180] Problem 5

problem number 1180

Added March 10, 2019.

Problem Chapter 4.8.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f_1(x) w_x +(f_2(x)+f_3(x) y^k)w_y = g(x) h(y) w \]

Mathematica

Failed

Maple

sol=()

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6.4.25.6 [1181] Problem 6

problem number 1181

Added March 10, 2019.

Problem Chapter 4.8.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f_1(x) g_1(y) w_x + f_2(x) g_2(y) w_y = h_1(x) h_2(y) w \]

Mathematica

Failed

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac{{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac{{\it g1} \left ( y \right ) }{{\it g2} \left ( y \right ) }}\,{\rm d}y \right ){{\rm e}^{\int ^{x}\!{\frac{{\it h1} \left ({\it \_f} \right ) }{{\it f1} \left ({\it \_f} \right ) }{\it h2} \left ( \RootOf \left ( \int \!{\frac{{\it f2} \left ({\it \_f} \right ) }{{\it f1} \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac{{\it g1} \left ({\it \_a} \right ) }{{\it g2} \left ({\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac{{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac{{\it g1} \left ( y \right ) }{{\it g2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \left ({\it g1} \left ( \RootOf \left ( \int \!{\frac{{\it f2} \left ({\it \_f} \right ) }{{\it f1} \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac{{\it g1} \left ({\it \_a} \right ) }{{\it g2} \left ({\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac{{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac{{\it g1} \left ( y \right ) }{{\it g2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \right ) ^{-1}}{d{\it \_f}}}}\] has RootOf

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6.4.25.7 [1182] Problem 7

problem number 1182

Added March 10, 2019.

Problem Chapter 4.8.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f_1(x) g_1(y) w_x + f_2(x) g_2(y) w_y = (h_1(x)+ h_2(y)) w \]

Mathematica

Failed

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac{{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac{{\it g1} \left ( y \right ) }{{\it g2} \left ( y \right ) }}\,{\rm d}y \right ){{\rm e}^{\int ^{x}\!{\frac{1}{{\it f1} \left ({\it \_f} \right ) } \left ({\it h1} \left ({\it \_f} \right ) +{\it h2} \left ( \RootOf \left ( \int \!{\frac{{\it f2} \left ({\it \_f} \right ) }{{\it f1} \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac{{\it g1} \left ({\it \_a} \right ) }{{\it g2} \left ({\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac{{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac{{\it g1} \left ( y \right ) }{{\it g2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \right ) \left ({\it g1} \left ( \RootOf \left ( \int \!{\frac{{\it f2} \left ({\it \_f} \right ) }{{\it f1} \left ({\it \_f} \right ) }}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac{{\it g1} \left ({\it \_a} \right ) }{{\it g2} \left ({\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac{{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac{{\it g1} \left ( y \right ) }{{\it g2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \right ) ^{-1}}{d{\it \_f}}}}\] has RootOf

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