6.9.1 2.1
6.9.1.1 [1919] Problem 1
problem number 1919
Added Jan 6, 2020.
Problem Chapter 9.2.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c w_z = (\alpha x + \beta ) w + p x + q \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+c*D[w[x,y,z],z]==(alpha*x+beta)*w[x,y,z]+p*x+q;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {x (\alpha x+2 \beta )}{2 a}} c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(\alpha x+\beta )^2}{2 a \alpha }} (\alpha q-\beta p) \text {erf}\left (\frac {\alpha x+\beta }{\sqrt {2} \sqrt {a} \sqrt {\alpha }}\right )}{\sqrt {a} \alpha ^{3/2}}-\frac {p}{\alpha }\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*diff(w(x,y,z),z)=(alpha*x+beta)*w(x,y,z)+p*x+q;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {-\frac {{\mathrm e}^{\frac {\left (\alpha x +\beta \right )^{2}}{2 a \alpha }} \sqrt {2}\, \sqrt {\pi }\, \left (-\alpha q +\beta p \right ) \operatorname {erf}\left (\frac {\sqrt {2}\, \left (\alpha x +\beta \right )}{2 a \sqrt {\frac {\alpha }{a}}}\right )}{2}+\left (f_{1} \left (\frac {y a -x b}{a}, \frac {z a -x c}{a}\right ) {\mathrm e}^{\frac {x \left (\alpha x +2 \beta \right )}{2 a}} \alpha -p \right ) a \sqrt {\frac {\alpha }{a}}}{\sqrt {\frac {\alpha }{a}}\, a \alpha }\]
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6.9.1.2 [1920] Problem 2
problem number 1920
Added Jan 6, 2020.
Problem Chapter 9.2.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a z w_y + b y w_z = (c x + k) w + p x + q \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x,y,z],x]+a*z*D[w[x,y,z],y]+b*y*D[w[x,y,z],z]==(c*x+k)*w[x,y,z]+p*x+q;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {1}{2} x (c x+2 k)} c_1\left (\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {b}},\frac {e^{-\sqrt {a} \sqrt {b} x} \left (\sqrt {a} z \left (e^{2 \sqrt {a} \sqrt {b} x}+1\right )-\sqrt {b} y \left (e^{2 \sqrt {a} \sqrt {b} x}-1\right )\right )}{2 \sqrt {a}}\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(c x+k)^2}{2 c}} \text {erf}\left (\frac {c x+k}{\sqrt {2} \sqrt {c}}\right ) (c q-k p)}{c^{3/2}}-\frac {p}{c}\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*z*diff(w(x,y,z),y)+ b*y*diff(w(x,y,z),z)=(c*x+k)*w(x,y,z)+p*x+q;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\left (f_{1} \left (\frac {a \,z^{2}-y^{2} b}{a}, \frac {x \sqrt {a b}-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right ) \sqrt {a b}+\int _{}^{y}\frac {\left (\left (p x +q \right ) \sqrt {a b}-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right ) p +\ln \left (\frac {a b \textit {\_a} +\sqrt {a b}\, \sqrt {\left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right ) a}}{\sqrt {a b}}\right ) p \right ) {\mathrm e}^{-\frac {\int \frac {\left (c x +k \right ) \sqrt {a b}-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right ) c +\ln \left (\frac {a b \textit {\_a} +\sqrt {a b}\, \sqrt {\left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right ) a}}{\sqrt {a b}}\right ) c}{\sqrt {\left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right ) a}}d \textit {\_a}}{\sqrt {a b}}}}{\sqrt {\left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right ) a}}d \textit {\_a} \right ) {\mathrm e}^{\frac {\int _{}^{y}\frac {\left (c x +k \right ) \sqrt {a b}-\ln \left (\frac {a b y +\sqrt {a^{2} z^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right ) c +\ln \left (\frac {a b \textit {\_a} +\sqrt {a b}\, \sqrt {\left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right ) a}}{\sqrt {a b}}\right ) c}{\sqrt {\left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right ) a}}d \textit {\_a}}{\sqrt {a b}}}}{\sqrt {a b}}\]
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6.9.1.3 [1921] Problem 3
problem number 1921
Added Jan 6, 2020.
Problem Chapter 9.2.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 x+a_0) w_y + (b_1 x+b_0) w_z = (c_1 x + c_0) w + s_1 x + s_0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x,y,z],x]+(a1*x+a0)*D[w[x,y,z],y]+(b1*x+b0)*D[w[x,y,z],z]==(c1*x+c0)*w[x,y,z]+s1*x+s0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {1}{2} x (2 \text {c0}+\text {c1} x)} c_1\left (-\text {a0} x-\frac {\text {a1} x^2}{2}+y,-\text {b0} x-\frac {\text {b1} x^2}{2}+z\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(\text {c0}+\text {c1} x)^2}{2 \text {c1}}} \text {erf}\left (\frac {\text {c0}+\text {c1} x}{\sqrt {2} \sqrt {\text {c1}}}\right ) (\text {c1} \text {s0}-\text {c0} \text {s1})}{\text {c1}^{3/2}}-\frac {\text {s1}}{\text {c1}}\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ (a__1*x+a__0)*diff(w(x,y,z),y)+ (b__1*x+b__0)*diff(w(x,y,z),z)=(c__1*x+c__0)*w(x,y,z)+s__1*x+s__0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {-2 s_{1} \sqrt {c_{1}}+2 f_{1} \left (-\frac {1}{2} a_{1} x^{2}-a_{0} x +y , -\frac {1}{2} b_{1} x^{2}-b_{0} x +z \right ) c_{1}^{{3}/{2}} {\mathrm e}^{\frac {x \left (c_{1} x +2 c_{0} \right )}{2}}+{\mathrm e}^{\frac {\left (c_{1} x +c_{0} \right )^{2}}{2 c_{1}}} \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \left (c_{1} x +c_{0} \right )}{2 \sqrt {c_{1}}}\right ) \left (-c_{0} s_{1} +c_{1} s_{0} \right )}{2 c_{1}^{{3}/{2}}}\]
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6.9.1.4 [1922] Problem 4
problem number 1922
Added Jan 6, 2020.
Problem Chapter 9.2.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (b_1 x+b_0) w_y + (c_1 y+c_0) w_z = a w + s_1 x + s_0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x,y,z],x]+(b1*x+b0)*D[w[x,y,z],y]+(c1*y+c0)*D[w[x,y,z],z]==a*w[x,y,z]+s1*x+s0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to -\frac {a^2 \left (-e^{a x}\right ) c_1\left (-\text {b0} x-\frac {\text {b1} x^2}{2}+y,\frac {1}{2} \text {b0} \text {c1} x^2+\frac {1}{3} \text {b1} \text {c1} x^3-\text {c0} x-\text {c1} x y+z\right )+a \text {s0}+a \text {s1} x+\text {s1}}{a^2}\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ (b__1*x+b__0)*diff(w(x,y,z),y)+ (c__1*x+c__0)*diff(w(x,y,z),z)=a*w(x,y,z)+s__1*x+s__0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {{\mathrm e}^{a x} f_{1} \left (-\frac {1}{2} b_{1} x^{2}-b_{0} x +y , -\frac {1}{2} c_{1} x^{2}-c_{0} x +z \right ) a^{2}+\left (-s_{1} x -s_{0} \right ) a -s_{1}}{a^{2}}\]
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6.9.1.5 [1923] Problem 5
problem number 1923
Added Jan 6, 2020.
Problem Chapter 9.2.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a y+k_1 x+k_0) w_y + (b z+n_1 x+n_0) w_z = (c_1 x+c_0) w + s_1 x + s_0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x,y,z],x]+(a*y+k1*x+k0)*D[w[x,y,z],y]+(b*z+n1*x+n0)*D[w[x,y,z],z]==(c1*x+c0)*w[x,y,z]+s1*x+s0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {1}{2} x (2 \text {c0}+\text {c1} x)} c_1\left (\frac {e^{-a x} \left (a^2 y+a (\text {k0}+\text {k1} x)+\text {k1}\right )}{a^2},\frac {e^{-b x} \left (b^2 z+b (\text {n0}+\text {n1} x)+\text {n1}\right )}{b^2}\right )+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {(\text {c0}+\text {c1} x)^2}{2 \text {c1}}} \text {erf}\left (\frac {\text {c0}+\text {c1} x}{\sqrt {2} \sqrt {\text {c1}}}\right ) (\text {c1} \text {s0}-\text {c0} \text {s1})}{\text {c1}^{3/2}}-\frac {\text {s1}}{\text {c1}}\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ (a*y+k__1*x+k__0)*diff(w(x,y,z),y)+ (b*z+n__1*x+n__0)*diff(w(x,y,z),z)=(c__1*x+c__0)*w(x,y,z)+s__1*x+s__0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {2 f_{1} \left (\frac {\left (y \,a^{2}+a \left (k_{1} x +k_{0} \right )+k_{1} \right ) {\mathrm e}^{-a x}}{a^{2}}, \frac {\left (z \,b^{2}+b \left (n_{1} x +n_{0} \right )+n_{1} \right ) {\mathrm e}^{-b x}}{b^{2}}\right ) c_{1}^{{3}/{2}} {\mathrm e}^{\frac {x \left (c_{1} x +2 c_{0} \right )}{2}}+{\mathrm e}^{\frac {\left (c_{1} x +c_{0} \right )^{2}}{2 c_{1}}} \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \left (c_{1} x +c_{0} \right )}{2 \sqrt {c_{1}}}\right ) \left (-c_{0} s_{1} +c_{1} s_{0} \right )-2 s_{1} \sqrt {c_{1}}}{2 c_{1}^{{3}/{2}}}\]
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6.9.1.6 [1924] Problem 6
problem number 1924
Added Jan 6, 2020.
Problem Chapter 9.2.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_2 y+a_1 x+a_0) w_y + (b_3 z+b_2 y+b_1 x+b_0) w_z = (c_3 z+c_2 y+c_1 x+c_0) w + s_3 z + s_2 y+s_1 x+s_0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x,y,z],x]+(a2*y+a1*x+a0)*D[w[x,y,z],y]+(b3*z+b2*y+b1*x+b0)*D[w[x,y,z],z]==(c3*z+c2*y+c1*x+c0)*w[x,y,z]+s3*z+s2*y+s1*x+s0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {\text {a2} \left (\text {a2} \left (2 (\text {b3} \text {c2}-\text {b2} \text {c3}) y \text {b3}^2+\text {a2} (\text {b3} (\text {b3} x (2 \text {c0}+\text {c1} x)+2 \text {c3} z)-2 \text {b0} \text {c3} (\text {b3} x-1)) \text {b3}+\text {a2} \text {b1} \text {c3} \left (2-\text {b3}^2 x^2\right )\right )-2 \text {a0} \text {b3} \left (\text {c2} (\text {a2} x-1) \text {b3}^2+\text {b2} \text {c3} (1-\text {a2} x) \text {b3}+\text {a2} \text {b2} \text {c3}\right )\right )-\text {a1} \left (\text {c2} \left (\text {a2}^2 x^2-2\right ) \text {b3}^3+\text {b2} \text {c3} \left (2-\text {a2}^2 x^2\right ) \text {b3}^2+2 \text {a2} \text {b2} \text {c3} \text {b3}+2 \text {a2}^2 \text {b2} \text {c3}\right )}{2 \text {a2}^3 \text {b3}^3}} \left (c_1\left (\frac {e^{-\text {a2} x} (\text {a2} x \text {a1}+\text {a1}+\text {a2} (\text {a0}+\text {a2} y))}{\text {a2}^2},\frac {e^{-((\text {a2}+\text {b3}) x)} \left (e^{\text {a2} x} (\text {b3} x \text {b1}+\text {b1}+\text {b3} (\text {b0}+\text {b3} z)) \text {a2}^3-\left (\text {a1} \text {b2} e^{\text {a2} x} (\text {b3} x+1)+\text {b3} \left (\text {a0} e^{\text {a2} x} \text {b2}+\text {b1} e^{\text {a2} x} (\text {b3} x+1)+\text {b3} \left (e^{\text {a2} x} \text {b0}+\text {b2} \left (e^{\text {a2} x}-e^{\text {b3} x}\right ) y+\text {b3} e^{\text {a2} x} z\right )\right )\right ) \text {a2}^2+\text {b2} \text {b3}^2 e^{\text {b3} x} (\text {a0}+\text {a1} x) \text {a2}+\text {a1} \text {b2} \text {b3}^2 e^{\text {b3} x}\right )}{\text {a2}^2 (\text {a2}-\text {b3}) \text {b3}^2}\right )+\int _1^x\frac {e^{\frac {e^{-((\text {a2}+\text {b3}) x)} \left (\text {a1} \left (e^{\text {a2} x} \left (\text {c2} e^{\text {b3} x} K[1]^2 \text {b3}^3+\text {b2} \text {c3} \left (2 e^{\text {b3} K[1]} (\text {b3} x+1)-\text {b3} e^{\text {b3} x} K[1] (\text {b3} K[1]+2)\right )\right ) \text {a2}^3-\text {b3}^3 e^{\text {b3} x} \left (-\text {b2} \text {c3} e^{\text {a2} x} K[1]^2+\text {c2} e^{\text {a2} x} (\text {b3} K[1]-2) K[1]+2 \text {c2} e^{\text {a2} K[1]} x\right ) \text {a2}^2+2 \text {b3}^3 e^{\text {b3} x} \left (\text {c2} e^{\text {a2} K[1]} (\text {b3} x-1)-\text {b3} \text {c2} e^{\text {a2} x} K[1]+\text {b2} \text {c3} \left (e^{\text {a2} x} K[1]-e^{\text {a2} K[1]} x\right )\right ) \text {a2}+2 \text {b3}^3 (\text {b3} \text {c2}-\text {b2} \text {c3}) e^{\text {b3} x+\text {a2} K[1]}\right )+\text {a2} \left (2 \text {a0} \text {b3} \left (e^{\text {a2} x} \left (\text {c2} e^{\text {b3} x} K[1] \text {b3}^2+\text {b2} \text {c3} \left (e^{\text {b3} K[1]}-\text {b3} e^{\text {b3} x} K[1]\right )\right ) \text {a2}^2-\text {b3}^2 e^{\text {b3} x} \left (\text {c2} \left (\text {b3} e^{\text {a2} x} K[1]+e^{\text {a2} K[1]}\right )-\text {b2} \text {c3} e^{\text {a2} x} K[1]\right ) \text {a2}+\text {b3}^2 (\text {b3} \text {c2}-\text {b2} \text {c3}) e^{\text {b3} x+\text {a2} K[1]}\right )+\text {a2} \left (-2 \text {a2}^3 e^{(\text {a2}+\text {b3}) x} x \text {b3}^3+2 (\text {b3} \text {c2}-\text {b2} \text {c3}) e^{\text {b3} x+\text {a2} K[1]} y \text {b3}^3+\text {a2} \left (\text {b1} \text {c3} e^{\text {a2} x} \left (2 e^{\text {b3} K[1]} (\text {b3} x+1)-\text {b3} e^{\text {b3} x} K[1] (\text {b3} K[1]+2)\right )+\text {b3} \left (2 e^{(\text {a2}+\text {b3}) x} x \text {b3}^3+e^{(\text {a2}+\text {b3}) x} K[1] (2 \text {c0}+\text {c1} K[1]) \text {b3}^2+\left (2 \text {c3} e^{\text {a2} x+\text {b3} K[1]} z-2 \text {c2} e^{\text {b3} x+\text {a2} K[1]} y\right ) \text {b3}+2 \text {b2} \text {c3} e^{\text {a2} x+\text {b3} K[1]} y+2 \text {b0} \text {c3} e^{\text {a2} x} \left (e^{\text {b3} K[1]}-\text {b3} e^{\text {b3} x} K[1]\right )\right )\right ) \text {b3}-\text {a2}^2 e^{\text {a2} x} \left (\text {b1} \text {c3} \left (2 e^{\text {b3} K[1]} (\text {b3} x+1)-\text {b3} e^{\text {b3} x} K[1] (\text {b3} K[1]+2)\right )+\text {b3} \left (2 \text {b0} \text {c3} \left (e^{\text {b3} K[1]}-\text {b3} e^{\text {b3} x} K[1]\right )+\text {b3} \left (2 \text {c3} e^{\text {b3} K[1]} z+\text {b3} e^{\text {b3} x} K[1] (2 \text {c0}+\text {c1} K[1])\right )\right )\right )\right )\right )\right )}{2 \text {a2}^3 (\text {a2}-\text {b3}) \text {b3}^3}} \left (\left (\left (e^{\text {b3} x+\text {a2} K[1]} \text {s2} y+e^{\text {a2} x+\text {b3} K[1]} \text {s3} z+e^{(\text {a2}+\text {b3}) x} (\text {s0}+\text {s1} K[1])\right ) \text {b3}^2-e^{\text {a2} x} \text {s3} \left (e^{\text {b3} x} \text {b0}-e^{\text {b3} K[1]} \text {b0}-\text {b1} e^{\text {b3} K[1]} x+\text {b1} e^{\text {b3} x} K[1]\right ) \text {b3}+\text {b1} \left (-e^{(\text {a2}+\text {b3}) x}+e^{\text {a2} x+\text {b3} K[1]}\right ) \text {s3}\right ) \text {a2}^3-\left (\left (e^{\text {b3} x+\text {a2} K[1]} \text {s2} y+e^{\text {a2} x+\text {b3} K[1]} \text {s3} z+e^{(\text {a2}+\text {b3}) x} (\text {s0}+\text {s1} K[1])\right ) \text {b3}^3+\left (\text {a0} e^{\text {b3} x} \left (e^{\text {a2} x}-e^{\text {a2} K[1]}\right ) \text {s2}-\text {a1} e^{\text {b3} x+\text {a2} K[1]} x \text {s2}+\text {a1} e^{(\text {a2}+\text {b3}) x} K[1] \text {s2}-\text {b0} e^{(\text {a2}+\text {b3}) x} \text {s3}+\text {b0} e^{\text {a2} x+\text {b3} K[1]} \text {s3}+\text {b1} e^{\text {a2} x+\text {b3} K[1]} \text {s3} x-\text {b2} e^{\text {b3} x+\text {a2} K[1]} \text {s3} y+\text {b2} e^{\text {a2} x+\text {b3} K[1]} \text {s3} y-\text {b1} e^{(\text {a2}+\text {b3}) x} \text {s3} K[1]\right ) \text {b3}^2-e^{\text {a2} x} \text {s3} \left (\text {b1} \left (e^{\text {b3} x}-e^{\text {b3} K[1]}\right )+\text {b2} \left (e^{\text {b3} x} \text {a0}-e^{\text {b3} K[1]} \text {a0}-\text {a1} e^{\text {b3} K[1]} x+\text {a1} e^{\text {b3} x} K[1]\right )\right ) \text {b3}-\text {a1} \text {b2} e^{\text {a2} x} \left (e^{\text {b3} x}-e^{\text {b3} K[1]}\right ) \text {s3}\right ) \text {a2}^2+\text {b3}^2 e^{\text {b3} x} \left (\text {a0} \left (e^{\text {a2} x}-e^{\text {a2} K[1]}\right ) (\text {b3} \text {s2}-\text {b2} \text {s3})+\text {a1} e^{\text {a2} K[1]} (-\text {b3} x \text {s2}+\text {s2}+\text {b2} \text {s3} x)-\text {a1} e^{\text {a2} x} (-\text {b3} K[1] \text {s2}+\text {s2}+\text {b2} \text {s3} K[1])\right ) \text {a2}+\text {a1} \text {b3}^2 e^{\text {b3} x} \left (e^{\text {a2} x}-e^{\text {a2} K[1]}\right ) (\text {b3} \text {s2}-\text {b2} \text {s3})\right )}{\text {a2}^2 (\text {a2}-\text {b3}) \text {b3}^2}dK[1]\right )\right \}\right \}\]
Maple ✗
restart;
local gamma;
pde := diff(w(x,y,z),x)+ (a__2*y+a__1*x+a__0)*diff(w(x,y,z),y)+ (b__3*z+b__2*y+b__1*x+b__0)*diff(w(x,y,z),z)=(c__3*z+c__2*y+c__1*x+c__0)*w(x,y,z)+s__3*z+s__2*y+s__1*x+s__0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
time expired
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6.9.1.7 [1925] Problem 7
problem number 1925
Added Jan 6, 2020.
Problem Chapter 9.2.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x w_x + b x w_y + c z w_z = (\alpha x+\beta ) w + p x+q \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*x*D[w[x,y,z],x]+b*x*D[w[x,y,z],y]+c*z*D[w[x,y,z],z]==(alpha*x+beta)*w[x,y,z]+p*x+q;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {e^{\frac {\alpha x}{a}} \left (-\left (\frac {\alpha x}{a}\right )^{\frac {\beta }{a}} \left (a p \Gamma \left (1-\frac {\beta }{a},\frac {\alpha x}{a}\right )+\alpha q \Gamma \left (-\frac {\beta }{a},\frac {\alpha x}{a}\right )\right )+a \alpha x^{\frac {\beta }{a}} c_1\left (y-\frac {b x}{a},z x^{-\frac {c}{a}}\right )\right )}{a \alpha }\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*x*diff(w(x,y,z),x)+ b*x*diff(w(x,y,z),y)+ c*z*diff(w(x,y,z),z)=(alpha*x+beta)*w(x,y,z)+p*x+q;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {-3 \left (a -\frac {\beta }{3}\right ) \left (x^{\frac {\beta }{a}} \left (a -\beta \right ) x^{\frac {-a -\beta }{a}}+\alpha \right ) q \alpha \,a^{2} {\mathrm e}^{\frac {\alpha x}{2 a}} \left (\frac {\alpha x}{a}\right )^{\frac {\beta }{2 a}} \operatorname {WhittakerM}\left (-\frac {1}{2}-\frac {\beta }{2 a}, 1-\frac {\beta }{2 a}, \frac {\alpha x}{a}\right )-3 \left (a -\frac {\beta }{3}\right ) x^{\frac {\beta }{a}} q \alpha a \left (a -\beta \right )^{2} {\mathrm e}^{\frac {\alpha x}{2 a}} \left (\frac {\alpha x}{a}\right )^{\frac {\beta }{2 a}} x^{\frac {-a -\beta }{a}} \operatorname {WhittakerM}\left (\frac {1}{2}-\frac {\beta }{2 a}, 1-\frac {\beta }{2 a}, \frac {\alpha x}{a}\right )+2 \left (2 x^{\frac {\beta }{a}} a^{2} {\mathrm e}^{\frac {\alpha x}{2 a}} \left (a -\frac {\beta }{2}\right )^{2} \left (\frac {\alpha x}{a}\right )^{\frac {\beta }{2 a}} p \,x^{\frac {-a -\beta }{a}} \operatorname {WhittakerM}\left (1-\frac {\beta }{2 a}, \frac {3}{2}-\frac {\beta }{2 a}, \frac {\alpha x}{a}\right )+\left (x^{\frac {\beta }{a}} \left (a -\frac {\beta }{2}\right ) x^{\frac {-a -\beta }{a}}+\frac {\alpha }{2}\right ) a^{3} {\mathrm e}^{\frac {\alpha x}{2 a}} \left (\frac {\alpha x}{a}\right )^{\frac {\beta }{2 a}} p \operatorname {WhittakerM}\left (-\frac {\beta }{2 a}, \frac {3}{2}-\frac {\beta }{2 a}, \frac {\alpha x}{a}\right )+3 \left (a -\frac {\beta }{3}\right ) f_{1} \left (y -\frac {b x}{a}, z \,x^{-\frac {c}{a}}\right ) x^{\frac {\beta }{a}} {\mathrm e}^{\frac {\alpha x}{a}} \alpha ^{2} \left (a -\beta \right ) \left (a -\frac {\beta }{2}\right )\right ) \beta \sqrt {\frac {\alpha x}{a}}}{\left (a -\beta \right ) \left (2 a -\beta \right ) \alpha ^{2} \left (3 a -\beta \right ) \sqrt {\frac {\alpha x}{a}}\, \beta }\]
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6.9.1.8 [1926] Problem 8
problem number 1926
Added Jan 6, 2020.
Problem Chapter 9.2.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x w_x + b y w_y + c z w_z = (\alpha x+\beta ) w + p x+q \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*x*D[w[x,y,z],x]+b*y*D[w[x,y,z],y]+c*z*D[w[x,y,z],z]==(alpha*x+beta)*w[x,y,z]+p*x+q;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {e^{\frac {\alpha x}{a}} \left (-\left (\frac {\alpha x}{a}\right )^{\frac {\beta }{a}} \left (a p \Gamma \left (1-\frac {\beta }{a},\frac {\alpha x}{a}\right )+\alpha q \Gamma \left (-\frac {\beta }{a},\frac {\alpha x}{a}\right )\right )+a \alpha x^{\frac {\beta }{a}} c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right )\right )}{a \alpha }\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := a*x*diff(w(x,y,z),x)+ b*y*diff(w(x,y,z),y)+ c*z*diff(w(x,y,z),z)=(alpha*x+beta)*w(x,y,z)+p*x+q;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {-3 \alpha \left (a -\frac {\beta }{3}\right ) \left (x^{\frac {\beta }{a}} \left (a -\beta \right ) x^{\frac {-a -\beta }{a}}+\alpha \right ) {\mathrm e}^{\frac {\alpha x}{2 a}} q \left (\frac {\alpha x}{a}\right )^{\frac {\beta }{2 a}} a^{2} \operatorname {WhittakerM}\left (-\frac {1}{2}-\frac {\beta }{2 a}, 1-\frac {\beta }{2 a}, \frac {\alpha x}{a}\right )-3 \alpha \left (a -\frac {\beta }{3}\right ) {\mathrm e}^{\frac {\alpha x}{2 a}} \left (a -\beta \right )^{2} q \,x^{\frac {\beta }{a}} \left (\frac {\alpha x}{a}\right )^{\frac {\beta }{2 a}} x^{\frac {-a -\beta }{a}} a \operatorname {WhittakerM}\left (\frac {1}{2}-\frac {\beta }{2 a}, 1-\frac {\beta }{2 a}, \frac {\alpha x}{a}\right )+2 \beta \sqrt {\frac {\alpha x}{a}}\, \left (2 \left (a -\frac {\beta }{2}\right )^{2} p \,{\mathrm e}^{\frac {\alpha x}{2 a}} x^{\frac {\beta }{a}} \left (\frac {\alpha x}{a}\right )^{\frac {\beta }{2 a}} x^{\frac {-a -\beta }{a}} a^{2} \operatorname {WhittakerM}\left (1-\frac {\beta }{2 a}, \frac {3}{2}-\frac {\beta }{2 a}, \frac {\alpha x}{a}\right )+p \,{\mathrm e}^{\frac {\alpha x}{2 a}} \left (x^{\frac {\beta }{a}} \left (a -\frac {\beta }{2}\right ) x^{\frac {-a -\beta }{a}}+\frac {\alpha }{2}\right ) \left (\frac {\alpha x}{a}\right )^{\frac {\beta }{2 a}} a^{3} \operatorname {WhittakerM}\left (-\frac {\beta }{2 a}, \frac {3}{2}-\frac {\beta }{2 a}, \frac {\alpha x}{a}\right )+3 \left (a -\frac {\beta }{2}\right ) f_{1} \left (y \,x^{-\frac {b}{a}}, z \,x^{-\frac {c}{a}}\right ) \alpha ^{2} \left (a -\frac {\beta }{3}\right ) {\mathrm e}^{\frac {\alpha x}{a}} \left (a -\beta \right ) x^{\frac {\beta }{a}}\right )}{\left (a -\beta \right ) \left (2 a -\beta \right ) \alpha ^{2} \left (3 a -\beta \right ) \sqrt {\frac {\alpha x}{a}}\, \beta }\]
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6.9.1.9 [1927] Problem 9
problem number 1927
Added Jan 6, 2020.
Problem Chapter 9.2.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + a z w_y + b y w_z = (c x+k) w + p x+q \]
Mathematica ✓
ClearAll["Global`*"];
pde = x*D[w[x,y,z],x]+a*z*D[w[x,y,z],y]+b*y*D[w[x,y,z],z]==(c*x+k)*w[x,y,z]+p*x+q;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {e^{c x} \left (-(c x)^k (p \Gamma (1-k,c x)+c q \Gamma (-k,c x))+c x^k c_1\left (i y \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {i \sqrt {a} z \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}},y \cosh \left (\sqrt {a} \sqrt {b} \log (x)\right )-\frac {\sqrt {a} z \sinh \left (\sqrt {a} \sqrt {b} \log (x)\right )}{\sqrt {b}}\right )\right )}{c}\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := x*diff(w(x,y,z),x)+ a*z*diff(w(x,y,z),y)+ b*y*diff(w(x,y,z),z)=(c*x+k)*w(x,y,z)+p*x+q;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\int _{}^{y}\frac {{\mathrm e}^{-\int \frac {c x \left (\sqrt {a b}\, y +a z \right )^{-\frac {\sqrt {a b}}{a b}} {\left (\frac {a b \textit {\_a} +\sqrt {a b}\, \sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right )}}{\sqrt {a b}}\right )}^{\frac {1}{\sqrt {a b}}}+k}{\sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right )}}d \textit {\_a}} \left (p x \left (\sqrt {a b}\, y +a z \right )^{-\frac {\sqrt {a b}}{a b}} {\left (\frac {a b \textit {\_a} +\sqrt {a b}\, \sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right )}}{\sqrt {a b}}\right )}^{\frac {1}{\sqrt {a b}}}+q \right )}{\sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right )}}d \textit {\_a} +f_{1} \left (z^{2}-\frac {b \,y^{2}}{a}, x \left (\sqrt {a b}\, y +a z \right )^{-\frac {\sqrt {a b}}{a b}}\right )\right ) {\mathrm e}^{\int _{}^{y}\frac {c x \left (\sqrt {a b}\, y +a z \right )^{-\frac {\sqrt {a b}}{a b}} {\left (\frac {a b \textit {\_a} +\sqrt {a b}\, \sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right )}}{\sqrt {a b}}\right )}^{\frac {1}{\sqrt {a b}}}+k}{\sqrt {a \left (\left (\textit {\_a}^{2}-y^{2}\right ) b +a \,z^{2}\right )}}d \textit {\_a}}\]
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