92 HFOPDE, chapter 3.8.2

 92.1 Problem 1
 92.2 Problem 2
 92.3 Problem 3
 92.4 Problem 4
 92.5 Problem 5
 92.6 Problem 6
 92.7 Problem 7

_______________________________________________________________________________________

92.1 Problem 1

problem number 791

Added Feb. 11, 2019.

Problem Chapter 3.8.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

92.2 Problem 2

problem number 792

Added Feb. 11, 2019.

Problem Chapter 3.8.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

92.3 Problem 3

problem number 793

Added Feb. 11, 2019.

Problem Chapter 3.8.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

92.4 Problem 4

problem number 794

Added Feb. 11, 2019.

Problem Chapter 3.8.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{DSolve}\left [f(x) w^{(1,0)}(x,y)+g(y) w^{(0,1)}(x,y)=\text{h1}(x)+\text{h2}(x),w(x,y),\{x,y\}\right ] \]

Maple

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

_______________________________________________________________________________________

92.5 Problem 5

problem number 795

Added Feb. 11, 2019.

Problem Chapter 3.8.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (y^{-k} \exp \left (-(1-k) \int _1^x \frac{\text{f2}(K[1])}{\text{f1}(K[1])} \, dK[1]\right ) \left (y^k \left (-\exp \left ((1-k) \int _1^x \frac{\text{f2}(K[1])}{\text{f1}(K[1])} \, dK[1]\right )\right ) \left (\int _1^x \frac{\text{f3}(K[2]) \exp \left (-(1-k) \text{Integrate}\left [\frac{\text{f2}(K[1])}{\text{f1}(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{\text{f1}(K[2])} \, dK[2]\right )+k y^k \exp \left ((1-k) \int _1^x \frac{\text{f2}(K[1])}{\text{f1}(K[1])} \, dK[1]\right ) \left (\int _1^x \frac{\text{f3}(K[2]) \exp \left (-(1-k) \text{Integrate}\left [\frac{\text{f2}(K[1])}{\text{f1}(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]\right )}{\text{f1}(K[2])} \, dK[2]\right )+y\right )\right )+\int _1^x \frac{g(K[3]) h(K[3])}{\text{f1}(K[3])} \, dK[3]\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int \!{\frac{g \left ( x \right ) h \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+{\it \_F1} \left ({y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac{{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x}}+k\int \!{\frac{{\it f3} \left ( x \right ) }{{\it f1} \left ( x \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac{{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x-\int \!{\frac{{\it f3} \left ( x \right ) }{{\it f1} \left ( x \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac{{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x \right ) \]

_______________________________________________________________________________________

92.6 Problem 6

problem number 796

Added Feb. 11, 2019.

Problem Chapter 3.8.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (y-\int _1^x \frac{\text{f2}(K[1]) \text{g2}(K[1])}{\text{f1}(K[1]) \text{g1}(K[1])} \, dK[1]\right )+\int _1^x \frac{\text{h1}(K[2]) \text{h2}(K[2])}{\text{f1}(K[2]) \text{g1}(K[2])} \, dK[2]\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int \!{\frac{{\it h1} \left ( x \right ){\it h2} \left ( x \right ) }{{\it f1} \left ( x \right ){\it g1} \left ( x \right ) }}\,{\rm d}x+{\it \_F1} \left ( -\int \!{\frac{{\it f2} \left ( x \right ){\it g2} \left ( x \right ) }{{\it f1} \left ( x \right ){\it g1} \left ( x \right ) }}\,{\rm d}x+y \right ) \]

_______________________________________________________________________________________

92.7 Problem 7

problem number 797

Added Feb. 11, 2019.

Problem Chapter 3.8.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{DSolve}\left [\text{f1}(x) \text{g1}(y) w^{(1,0)}(x,y)+\text{f2}(x) \text{g2}(y) w^{(0,1)}(x,y)=\text{h1}(x)+\text{h2}(x),w(x,y),\{x,y\}\right ] \]

Maple

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