6.3.2 2.1

6.3.2.1 [846] Problem 1

problem number 846

Added Feb. 9, 2019.

Problem Chapter 3.2.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )+\frac {c x}{a}\right \}\right \}\]

Maple ✓

restart; 
pde := a* diff(w(x,y),x)+b*diff(w(x,y),y) = c; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\frac {cx}{a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]

____________________________________________________________________________________

6.3.2.2 [847] Problem 2

problem number 847

Added Feb. 9, 2019.

Problem Chapter 3.2.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \alpha x+ \beta y + \gamma \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == alpha*x + beta*y + gamma; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )+\frac {x (a (\alpha x+2 \beta y+2 \gamma )-b \beta x)}{2 a^2}\right \}\right \}\]

Maple ✓

restart; 
pde := a* diff(w(x,y),x)+b*diff(w(x,y),y) = alpha*x+beta*y+gamma; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) =-1/2\,{\frac { \left ( -a\alpha +b\beta \right ) {x}^{2}}{{a}^{2}}}+1/8\,{\frac { \left ( 8\,\beta \,y+1 \right ) x}{a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]

____________________________________________________________________________________

6.3.2.3 [848] Problem 3

problem number 848

Added Feb. 9, 2019.

Problem Chapter 3.2.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b w_y = \alpha x+ \beta y + \gamma \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x*D[w[x, y], x] + b*D[w[x, y], y] == alpha*x + beta*y + gamma; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b \log (x)}{a}\right )+\frac {2 a \alpha x+2 a \log (x) (\beta y+\gamma )-b \beta \log ^2(x)}{2 a^2}\right \}\right \}\]

Maple ✓

restart; 
pde := a*x* diff(w(x,y),x)+b*diff(w(x,y),y) = alpha*x+beta*y+gamma; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) =1/8\,{\frac {1}{{a}^{2}} \left ( 8\,{\it \_F1} \left ( {\frac {ya-b\ln \left ( x \right ) }{a}} \right ) {a}^{2}-4\,b\beta \, \left ( \ln \left ( x \right ) \right ) ^{2}+a \left ( 8\,\beta \,y+1 \right ) \ln \left ( x \right ) +8\,\alpha \,xa \right ) }\]

____________________________________________________________________________________

6.3.2.4 [849] Problem 4

problem number 849

Added Feb. 9, 2019.

Problem Chapter 3.2.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b x w_y = c \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x*D[w[x, y], x] + b*x*D[w[x, y], y] == c; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )+\frac {c \log (x)}{a}\right \}\right \}\]

Maple ✓

restart; 
pde := a*x* diff(w(x,y),x)+b*x*diff(w(x,y),y) = c; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\frac {1}{a} \left ( \ln \left ( x \right ) c+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) a \right ) }\]

____________________________________________________________________________________

6.3.2.5 [850] Problem 5

problem number 850

Added Feb. 9, 2019.

Problem Chapter 3.2.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x +b) w_x + (c y +d) w_y = \alpha x+ \beta y + \gamma \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (a*x + b)*D[w[x, y], x] + (c*y + d)*D[w[x, y], y] == alpha*x + beta*y + gamma; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {(c y+d) (a x+b)^{-\frac {c}{a}}}{c}\right )+\frac {\log (a x+b) (-a \beta d+a c \gamma -\alpha b c)}{a^2 c}+\frac {\alpha x}{a}+\frac {\beta (c y+d)}{c^2}\right \}\right \}\]

Maple ✓

restart; 
pde := (a*x+b)* diff(w(x,y),x)+(c*y+d)*diff(w(x,y),y) = alpha*x+beta*y+gamma; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) =1/8\,{\frac {1}{{a}^{2}{c}^{2}} \left ( 8\,{\it \_F1} \left ( {\frac {cy+d}{c} \left ( ax+b \right ) ^{-{\frac {c}{a}}}} \right ) {a}^{2}{c}^{2}+c \left ( \left ( -8\,\beta \,d+c \right ) a-8\,\alpha \,bc \right ) \ln \left ( ax+b \right ) +8\, \left ( \left ( cy+d \right ) \beta \,a+{c}^{2}x\alpha \right ) a \right ) }\]

____________________________________________________________________________________

6.3.2.6 [851] Problem 6

problem number 851

Added Feb. 9, 2019.

Problem Chapter 3.2.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y w_x + b w_y = \alpha x+ \beta y + \gamma \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*y*D[w[x, y], x] + b*D[w[x, y], y] == alpha*x + beta*y + gamma; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\begin {align*} & \left \{w(x,y)\to c_1\left (\frac {y^2}{2}-\frac {b x}{a}\right )-\frac {\alpha \left (a y^2\right )^{3/2}}{3 \sqrt {a} b^2}+\frac {\sqrt {a y^2} (\alpha x+\gamma )}{\sqrt {a} b}+\frac {\beta x}{a}\right \}\\& \left \{w(x,y)\to c_1\left (\frac {y^2}{2}-\frac {b x}{a}\right )+\frac {\alpha \left (a y^2\right )^{3/2}}{3 \sqrt {a} b^2}-\frac {\sqrt {a y^2} (\alpha x+\gamma )}{\sqrt {a} b}+\frac {\beta x}{a}\right \}\\ \end {align*}

Maple ✓

restart; 
pde := a*y* diff(w(x,y),x)+b*diff(w(x,y),y) = alpha*x+beta*y+gamma; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) =1/24\,{\frac {1}{{b}^{2}{a}^{2}} \left ( 24\,{\it \_F1} \left ( {\frac {{y}^{2}a-2\,bx}{a}} \right ) {b}^{2}{a}^{2}+4\,\alpha \, \left ( {y}^{2}{a}^{2} \right ) ^{3/2}-12\, \left ( \left ( a{y}^{2}\alpha -2\,\alpha \,bx-b/4 \right ) \sqrt {{y}^{2}{a}^{2}}-2\,{b}^{2}x\beta \right ) a \right ) }\]

____________________________________________________________________________________

6.3.2.7 [852] Problem 7

problem number 852

Added Feb. 9, 2019.

Problem Chapter 3.2.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*y*D[w[x, y], x] + b*x*D[w[x, y], y] == c; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\begin {align*} & \left \{w(x,y)\to c_1\left (\frac {a y^2-b x^2}{2 a}\right )-\frac {c \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a y^2}}\right )}{\sqrt {a} \sqrt {b}}\right \}\\& \left \{w(x,y)\to c_1\left (\frac {a y^2-b x^2}{2 a}\right )+\frac {c \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a y^2}}\right )}{\sqrt {a} \sqrt {b}}\right \}\\ \end {align*}

Maple ✓

restart; 
pde := a*y* diff(w(x,y),x)+b*x*diff(w(x,y),y) = c; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\frac {1}{\sqrt {ab}} \left ( \ln \left ( {\frac {axb}{\sqrt {ab}}}+\sqrt {{y}^{2}{a}^{2}} \right ) c+{\it \_F1} \left ( {\frac {{y}^{2}a-b{x}^{2}}{a}} \right ) \sqrt {ab} \right ) }\]

____________________________________________________________________________________

6.3.2.8 [853] Problem 8

problem number 853

Added Feb. 9, 2019.

Problem Chapter 3.2.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y w_x + b x w_y = c x+ k y \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*y*D[w[x, y], x] + b*x*D[w[x, y], y] == c*x + k*y; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\begin {align*} & \left \{w(x,y)\to c_1\left (\frac {a y^2-b x^2}{2 a}\right )-\frac {c \sqrt {a y^2}}{\sqrt {a} b}+\frac {k x}{a}\right \}\\& \left \{w(x,y)\to c_1\left (\frac {a y^2-b x^2}{2 a}\right )+\frac {c \sqrt {a y^2}}{\sqrt {a} b}+\frac {k x}{a}\right \}\\ \end {align*}

Maple ✓

restart; 
pde := a*y* diff(w(x,y),x)+b*x*diff(w(x,y),y) = c*x+k*y; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\frac {cy}{b}}+{\frac {kx}{a}}+{\it \_F1} \left ( {\frac {{y}^{2}a-b{x}^{2}}{a}} \right ) \]

____________________________________________________________________________________