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'));
 

\[w \left (x , y\right ) = \frac {c x}{a}+f_{1} \left (\frac {y a -b x}{a}\right )\]

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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'));
 

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

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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'));
 

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

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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'));
 

\[w \left (x , y\right ) = \frac {f_{1} \left (\frac {y a -b x}{a}\right ) a +c \ln \left (x \right )}{a}\]

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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'));
 

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

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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'));
 

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

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

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

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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'));
 

\[w \left (x , y\right ) = \frac {y c}{b}+\frac {k x}{a}+f_{1} \left (\frac {y^{2} a -b \,x^{2}}{a}\right )\]

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