6.3.2 2.1

6.3.2.1 [845] Problem 1

problem number 845

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]];
 

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

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'));
 

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

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6.3.2.2 [846] Problem 2

problem number 846

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]];
 

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

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'));
 

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

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6.3.2.3 [847] Problem 3

problem number 847

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]];
 

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

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'));
 

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

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6.3.2.4 [848] Problem 4

problem number 848

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]];
 

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

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'));
 

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

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6.3.2.5 [849] Problem 5

problem number 849

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'));
 

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

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6.3.2.6 [850] Problem 6

problem number 850

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'));
 

\[w \left ( x,y \right ) ={\frac {1}{6\,{a}^{2}{b}^{2}} \left ( 6\,{\it \_F1} \left ( {\frac {{y}^{2}a-2\,bx}{a}} \right ) {a}^{2}{b}^{2}-3\,a \left ( a{y}^{2}\alpha -2\,b \left ( \alpha \,x+\gamma \right ) \right ) \sqrt {{a}^{2}{y}^{2}}+6\,\beta \,xa{b}^{2}+ \left ( {a}^{2}{y}^{2} \right ) ^{{\frac {3}{2}}}\alpha \right ) }\]

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6.3.2.7 [851] Problem 7

problem number 851

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

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

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6.3.2.8 [852] Problem 8

problem number 852

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'));
 

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

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