Added March 10, 2019.
Problem Chapter 5.2.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + d \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x, y] + d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {d}{c}+e^{\frac {c x}{a}} c_1\left (y-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+b*diff(w(x,y),y) = c*w(x,y)+d; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {c \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {c x}{a}}-d}{c}\]
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Added March 10, 2019.
Problem Chapter 5.2.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (x-a) w_x + (y-b) w_y = w - c \]
Mathematica ✓
ClearAll["Global`*"]; pde = (x - a)*D[w[x, y], x] + (y - b)*D[w[x, y], y] == w[x, y] - c; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c+(x-a) c_1\left (\frac {b-y}{a-x}\right )\right \}\right \}\]
Maple ✓
restart; pde := (x-a)*diff(w(x,y),x)+(y-b)*diff(w(x,y),y) = w(x,y)-c; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = c +\left (a -x \right ) \textit {\_F1} \left (\frac {-b +y}{a -x}\right )\]
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Added March 10, 2019.
Problem Chapter 5.2.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x+b) w_x + (c x+d) w_y = \alpha w + \beta \]
Mathematica ✓
ClearAll["Global`*"]; pde = (a*x + b)*D[w[x, y], x] + (c*x + d)*D[w[x, y], y] == alpha*w[x, y] + beta; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {\beta }{\alpha }+(a x+b)^{\frac {\alpha }{a}} c_1\left (\frac {(b c-a d) \log (a x+b)+a (a y-c x)}{a^2}\right )\right \}\right \}\]
Maple ✓
restart; pde := (a*x+b)*diff(w(x,y),x)+ (c*x+d)*diff(w(x,y),y) = alpha*w(x,y)+beta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\alpha \left (a x +b \right )^{\frac {\alpha }{a}} \textit {\_F1} \left (\frac {a^{2} y -a c x -a d \ln \left (a x +b \right )+b c \ln \left (a x +b \right )}{a^{2}}\right )-\beta }{\alpha }\]
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Added March 10, 2019.
Problem Chapter 5.2.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x+b) w_x + (c y+d) w_y = \alpha w + \beta \]
Mathematica ✓
ClearAll["Global`*"]; pde = (a*x + b)*D[w[x, y], x] + (c*y + d)*D[w[x, y], y] == alpha*w[x, y] + beta; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {\beta }{\alpha }+(a x+b)^{\frac {\alpha }{a}} c_1\left (\frac {(c y+d) (a x+b)^{-\frac {c}{a}}}{c}\right )\right \}\right \}\]
Maple ✓
restart; pde := (a*x+b)*diff(w(x,y),x)+ (c*y+d)*diff(w(x,y),y) = alpha*w(x,y)+beta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\alpha \left (a x +b \right )^{\frac {\alpha }{a}} \textit {\_F1} \left (\frac {\left (c y +d \right ) \left (a x +b \right )^{-\frac {c}{a}}}{c}\right )-\beta }{\alpha }\]
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Added March 10, 2019.
Problem Chapter 5.2.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x+b) w_x + (c y+d) w_y = \alpha w + \beta y+\gamma x \]
Mathematica ✓
ClearAll["Global`*"]; pde = (a*x + b)*D[w[x, y], x] + (c*y + d)*D[w[x, y], y] == alpha*w[x, y] + beta*y + gamma*x; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {\alpha (a-\alpha ) (\alpha -c) (a x+b)^{\frac {\alpha }{a}} c_1\left (\frac {(c y+d) (a x+b)^{-\frac {c}{a}}}{c}\right )-a \beta (\alpha y+d)+\alpha ^2 \beta y+\alpha ^2 \gamma x+\alpha b \gamma +\alpha \beta d-\alpha c \gamma x-b c \gamma }{\alpha (\alpha -a) (\alpha -c)}\right \}\right \}\]
Maple ✓
restart; pde := (a*x+b)*diff(w(x,y),x)+ (c*y+d)*diff(w(x,y),y) = alpha*w(x,y)+beta*y+gamma*x; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {a \beta d +\left (-\alpha +c \right ) \left (a -\alpha \right ) \alpha \left (a x +b \right )^{\frac {\alpha }{a}} \textit {\_F1} \left (\frac {\left (c y +d \right ) \left (a x +b \right )^{-\frac {c}{a}}}{c}\right )+\left (-\beta y -\gamma x \right ) \alpha ^{2}+\gamma b c +\left (a \beta y -\beta d +\gamma c x -\gamma b \right ) \alpha }{\left (a -\alpha \right ) \left (-\alpha +c \right ) \alpha }\]
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Added March 10, 2019.
Problem Chapter 5.2.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x+b) w_x + (c x+d y) w_y = \alpha w + \beta \]
Mathematica ✓
ClearAll["Global`*"]; pde = (a*x + b)*D[w[x, y], x] + (c*x + d*y)*D[w[x, y], y] == alpha*w[x, y] + beta; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {\beta }{\alpha }+(a x+b)^{\frac {\alpha }{a}} c_1\left (\frac {(a x+b)^{-\frac {d}{a}} (-d (-a y+c x+d y)-b c)}{d (a-d)}\right )\right \}\right \}\]
Maple ✓
restart; pde := (a*x+b)*diff(w(x,y),x)+ (c*x+d*y)*diff(w(x,y),y) = alpha*w(x,y)+beta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\alpha \left (a x +b \right )^{\frac {\alpha }{a}} \textit {\_F1} \left (\frac {\left (d y a -c x d -d^{2} y -b c \right ) \left (a x +b \right )^{-\frac {d}{a}}}{\left (a -d \right ) d}\right )-\beta }{\alpha }\]
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Added March 10, 2019.
Problem Chapter 5.2.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a_1 x+a_0) w_x + (b_2 y+b_1 x+b_0) w_y = (c_2 y+c_1 x+c_0) w + k_2 y+k_1 x+k_0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = (a1*x + a0)*D[w[x, y], x] + (b2*y + b1*x + b0)*D[w[x, y], y] == (c2*y + c1*x + c0)*w[x, y] + k2*y + k1*x + k0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to (\text {a0}+\text {a1} x)^{\frac {\text {a0} \text {b1} \text {c2}-\text {a0} \text {b2} \text {c1}-\text {a1} \text {b0} \text {c2}+\text {a1} \text {b2} \text {c0}}{\text {a1}^2 \text {b2}}} \exp \left (\frac {-\text {a1} (\text {a0} \text {b1} \text {c2}+\text {b2} (\text {b0} \text {c2}+\text {b1} \text {c2} x-\text {b2} \text {c1} x+\text {b2} \text {c2} y))+\text {a1}^2 \text {c2} (\text {b0}+\text {b2} y)+\text {b2}^2 x (\text {b1} \text {c2}-\text {b2} \text {c1})}{\text {a1} \text {b2}^2 (\text {a1}-\text {b2})}\right ) \left (\int _1^x\frac {\exp \left (-\frac {\frac {(\text {a1} \text {c1}-\text {b2} \text {c1}+\text {b1} \text {c2}) K[1]}{\text {a1}}-\frac {\text {c2} (\text {a0}+\text {a1} x)^{-\frac {\text {b2}}{\text {a1}}} (\text {a0} \text {b1}-\text {a1} (\text {b0}+\text {b2} y)+\text {b2} (\text {b0}+\text {b1} x+\text {b2} y)) (\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}}{\text {b2}^2}}{\text {a1}-\text {b2}}\right ) (\text {a0}+\text {a1} x)^{-\frac {\text {b2}}{\text {a1}}} (\text {a0}+\text {a1} K[1])^{\frac {-\text {a1} \text {b2} \text {c0}+\text {a0} \text {b2} \text {c1}+\text {a1} \text {b0} \text {c2}-\text {a0} \text {b1} \text {c2}}{\text {a1}^2 \text {b2}}-1} \left (-\left (\left (\text {k0} (\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}+\text {k1} K[1] (\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}+\text {k2} y (\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}\right ) \text {b2}^2\right )+\text {a1} \left (\text {k0} (\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}+\text {k1} K[1] (\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}+\text {k2} y (\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}\right ) \text {b2}+\text {k2} \left (\text {b1} K[1] (\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}-\text {b1} x (\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}+\text {b0} \left ((\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}-(\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}\right )\right ) \text {b2}+\text {a0} \text {b1} \text {k2} \left ((\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}-(\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}\right )+\text {a1} \text {b0} \text {k2} \left ((\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}-(\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}\right )\right )}{(\text {a1}-\text {b2}) \text {b2}}dK[1]+c_1\left (\frac {(\text {a0}+\text {a1} x)^{-\frac {\text {b2}}{\text {a1}}} (-\text {a0} \text {b1}+\text {a1} (\text {b0}+\text {b2} y)-\text {b2} (\text {b0}+\text {b1} x+\text {b2} y))}{\text {b2} (\text {a1}-\text {b2})}\right )\right )\right \}\right \}\]
Maple ✗
restart; pde := (a1*x+a0)*diff(w(x,y),x)+ (b2*y+b1*x+b0)*diff(w(x,y),y) = (c2*y+c1*x+c0)*w(x,y)+k2*y+k1*x+k0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
time expired
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Added March 10, 2019.
Problem Chapter 5.2.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y w_x + (b_1 x+b_0) w_y = (c_1 x+c_0) w + s_1 x+s0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*y*D[w[x, y], x] + (b1*x + b0)*D[w[x, y], y] == (c1*x + c0)*w[x, y] + s1*x + s0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\begin {align*} & \left \{w(x,y)\to \exp \left (\frac {(\text {b0} \text {c1}-\text {b1} \text {c0}) \tanh ^{-1}\left (\frac {\text {b0}+\text {b1} x}{\sqrt {\text {b1}} \sqrt {a y^2}}\right )-\sqrt {\text {b1}} \text {c1} \sqrt {a y^2}}{\sqrt {a} \text {b1}^{3/2}}\right ) \left (\int _1^x-\frac {\exp \left (\frac {\frac {\sqrt {-\text {b1} x^2-2 \text {b0} x+a y^2+\text {b1} K[1]^2+2 \text {b0} K[1]} \text {c1}}{\text {b1}}+\frac {(\text {b1} \text {c0}-\text {b0} \text {c1}) \tanh ^{-1}\left (\frac {\text {b0}+\text {b1} K[1]}{\sqrt {\text {b1}} \sqrt {-\text {b1} x^2-2 \text {b0} x+a y^2+\text {b1} K[1]^2+2 \text {b0} K[1]}}\right )}{\text {b1}^{3/2}}}{\sqrt {a}}\right ) (\text {s0}+\text {s1} K[1])}{\sqrt {a} \sqrt {-\text {b1} x^2-2 \text {b0} x+a y^2+\text {b1} K[1]^2+2 \text {b0} K[1]}}dK[1]+c_1\left (-\frac {-a y^2+2 \text {b0} x+\text {b1} x^2}{2 a}\right )\right )\right \}\\& \left \{w(x,y)\to \exp \left (\frac {(\text {b1} \text {c0}-\text {b0} \text {c1}) \tanh ^{-1}\left (\frac {\text {b0}+\text {b1} x}{\sqrt {\text {b1}} \sqrt {a y^2}}\right )+\sqrt {\text {b1}} \text {c1} \sqrt {a y^2}}{\sqrt {a} \text {b1}^{3/2}}\right ) \left (\int _1^x\frac {\exp \left (-\frac {\frac {\sqrt {-\text {b1} x^2-2 \text {b0} x+a y^2+\text {b1} K[2]^2+2 \text {b0} K[2]} \text {c1}}{\text {b1}}+\frac {(\text {b1} \text {c0}-\text {b0} \text {c1}) \tanh ^{-1}\left (\frac {\text {b0}+\text {b1} K[2]}{\sqrt {\text {b1}} \sqrt {-\text {b1} x^2-2 \text {b0} x+a y^2+\text {b1} K[2]^2+2 \text {b0} K[2]}}\right )}{\text {b1}^{3/2}}}{\sqrt {a}}\right ) (\text {s0}+\text {s1} K[2])}{\sqrt {a} \sqrt {-\text {b1} x^2-2 \text {b0} x+a y^2+\text {b1} K[2]^2+2 \text {b0} K[2]}}dK[2]+c_1\left (-\frac {-a y^2+2 \text {b0} x+\text {b1} x^2}{2 a}\right )\right )\right \}\\ \end {align*}
Maple ✓
restart; pde := a*y*diff(w(x,y),x)+ (b1*x+b0)*diff(w(x,y),y) = (c1*x+c0)*w(x,y)+s1*x+s0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\textit {\_a} \mathit {s1} +\mathit {s0} \right ) \left (\frac {\left (\textit {\_a} \mathit {b1} +\mathit {b0} \right ) a +\sqrt {\left (a \,y^{2}+\left (2 \textit {\_a} -2 x \right ) \mathit {b0} +\left (\textit {\_a}^{2}-x^{2}\right ) \mathit {b1} \right ) a}\, \sqrt {a \mathit {b1}}}{\sqrt {a \mathit {b1}}}\right )^{\frac {\mathit {b0} \mathit {c1} -\mathit {b1} \mathit {c0}}{\sqrt {a \mathit {b1}}\, \mathit {b1}}} {\mathrm e}^{-\frac {\sqrt {\left (a \,y^{2}+\left (2 \textit {\_a} -2 x \right ) \mathit {b0} +\left (\textit {\_a}^{2}-x^{2}\right ) \mathit {b1} \right ) a}\, \mathit {c1}}{a \mathit {b1}}}}{\sqrt {\left (a \,y^{2}+\left (2 \textit {\_a} -2 x \right ) \mathit {b0} +\left (\textit {\_a}^{2}-x^{2}\right ) \mathit {b1} \right ) a}}d \textit {\_a} +\textit {\_F1} \left (\frac {a \,y^{2}-\mathit {b1} \,x^{2}-2 \mathit {b0} x}{a}\right )\right ) \left (\frac {\left (\mathit {b1} x +\mathit {b0} \right ) a}{\sqrt {a \mathit {b1}}}+\sqrt {a^{2} y^{2}}\right )^{-\frac {\mathit {b0} \mathit {c1} -\mathit {b1} \mathit {c0}}{\sqrt {a \mathit {b1}}\, \mathit {b1}}} {\mathrm e}^{\frac {\sqrt {a^{2} y^{2}}\, \mathit {c1}}{a \mathit {b1}}}\]
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