Added June 28, 2019.
Problem Chapter 8.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c w_z = (\lambda x^2 +\beta y^2+\gamma z^2+\delta ) w \]
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^2+beta*y^2+gamma*z^2+delta)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\frac {c x}{a}\right ) \exp \left (\frac {1}{3} \left (\frac {\alpha x^3+3 \delta x}{a}+\frac {\beta y^3}{b}+\frac {\gamma z^3}{c}\right )\right )\right \}\right \}\]
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^2+beta*y^2+gamma*z^2+delta)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},{\frac {za-cx}{a}} \right ) {{\rm e}^{1/3\,{\frac { \left ( \left ( \alpha \,{x}^{2}+3\,\beta \,{y}^{2}+3/8\,{z}^{2}+3\,\delta \right ) {a}^{2}-3\,x \left ( by\beta +1/8\,cz \right ) a+{x}^{2} \left ( {b}^{2}\beta +1/8\,{c}^{2} \right ) \right ) x}{{a}^{3}}}}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 x^2+a_0) w_y + (b_1 x^2+b_0) w_z = (\lambda x +\beta y+\gamma z+\delta ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a1*x^2+a0)*D[w[x, y,z], y] +(b1*x^2+b0)*D[w[x,y,z],z]== (alpha*x+beta*y+gamma*z+delta)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\text {a0} x-\frac {\text {a1} x^3}{3}+y,-\text {b0} x-\frac {\text {b1} x^3}{3}+z\right ) \exp \left (-\frac {1}{4} x \left (2 \text {a0} \beta x+\text {a1} \beta x^3-2 \alpha x+2 \text {b0} \gamma x+\text {b1} \gamma x^3-4 \beta y-4 \delta -4 \gamma z\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+(a1*x^2+a0)*diff(w(x,y,z),y)+(b1*x^2+b0)*diff(w(x,y,z),z)= (alpha*x+beta*y+gamma*z+delta)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -1/3\,{\it a1}\,{x}^{3}-{\it a0}\,x+y,-1/3\,{\it b1}\,{x}^{3}-{\it b0}\,x+z \right ) {{\rm e}^{-1/4\, \left ( \left ( \beta \,{\it a1}+{\it b1}/8 \right ) {x}^{3}+ \left ( 2\,\beta \,{\it a0}+{\it b0}/4-2\,\alpha \right ) x-4\,\beta \,y-4\,\delta -z/2 \right ) x}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a y+ k_1 x^2+k_0) w_y + (b z+s_1 x^2+s_0) w_z = (c_1 x^2+c_0) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a*y+k1*x^2+k0)*D[w[x, y,z], y] +(b*z+s1*x^2+s0)*D[w[x,y,z],z]== (c1*x^2+c0)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\text {c0} x+\frac {\text {c1} x^3}{3}} c_1\left (\frac {e^{-a x} \left (a^3 y+a^2 \left (\text {k0}+\text {k1} x^2\right )+2 a \text {k1} x+2 \text {k1}\right )}{a^3},\frac {e^{-b x} \left (b^3 z+b^2 \left (\text {s0}+\text {s1} x^2\right )+2 b \text {s1} x+2 \text {s1}\right )}{b^3}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+(a*y+k1*x^2+k0)*diff(w(x,y,z),y)+(b*z+s1*x^2+s0)*diff(w(x,y,z),z)= (c1*x^2+c0)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {{{\rm e}^{-ax}} \left ( {a}^{3}y+ \left ( {\it k1}\,{x}^{2}+{\it k0} \right ) {a}^{2}+2\,{\it k1}\,xa+2\,{\it k1} \right ) }{{a}^{3}}},{\frac {{{\rm e}^{-bx}} \left ( z{b}^{3}+ \left ( {\it s1}\,{x}^{2}+{\it s0} \right ) {b}^{2}+2\,{\it s1}\,xb+2\,{\it s1} \right ) }{{b}^{3}}} \right ) {{\rm e}^{1/3\,x \left ( {\it c1}\,{x}^{2}+3\,{\it c0} \right ) }}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_2 x y+ a_1 x^2+a_0) w_y + (b_2 x y+b_1 x^2+b_0) w_z = (c_2 y+c_1 z+c_0 x+s) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a1*x*y+a1*x^2+a0)*D[w[x, y,z], y] +(b2*x*y+b1*x^2+b0)*D[w[x,y,z],z]== (c2*x+c1*z+c0*x+s)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\text {a0} \text {b2} x}{\text {a1}}-\frac {\text {b2} y}{\text {a1}}-\text {b0} x-\frac {\text {b1} x^3}{3}+\frac {\text {b2} x^3}{3}+z,e^{-\frac {\text {a1} x^2}{2}} (x+y)-\frac {\sqrt {\frac {\pi }{2}} (\text {a0}+1) \text {Erf}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right )}{\sqrt {\text {a1}}}\right ) \exp \left (\frac {2 (\text {a0}+1) \text {a1} \text {b2} \text {c1} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {\text {a1} x^2}{2}\right )+\text {a1} x \left (2 \text {b2} \text {c1} ((\text {a0}-1) x-2 y)+\text {a1} \left (-2 \text {b0} \text {c1} x-\text {b1} \text {c1} x^3+\text {b2} \text {c1} x^3+2 \text {c0} x+4 \text {c1} z+2 \text {c2} x+4 s\right )\right )+2 \text {b2} \text {c1} e^{-\frac {\text {a1} x^2}{2}} \text {Erfi}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right ) \left (\sqrt {2 \pi } \sqrt {\text {a1}} (x+y)-\pi (\text {a0}+1) e^{\frac {\text {a1} x^2}{2}} \text {Erf}\left (\frac {\sqrt {\text {a1}} x}{\sqrt {2}}\right )\right )}{4 \text {a1}^2}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+(a1*x*y+a1*x^2+a0)*diff(w(x,y,z),y)+(b2*x*y+b1*x^2+b0)*diff(w(x,y,z),z)= (c2*x+c1*z+c0*x+s)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {\sqrt {{\it a1}} \left ( x+y \right ) {{\rm e}^{-1/2\,{\it a1}\,{x}^{2}}}-1/2\,\sqrt {2}\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) \sqrt {\pi } \left ( {\it a0}+1 \right ) }{\sqrt {{\it a1}}}},-1/3\,{\frac {1}{{\it a1}} \left ( 3\,{\it b2}\, \left ( \sqrt {{\frac {{\it a1}}{\pi }}} \left ( x+y \right ) {{\rm e}^{-1/2\,{\it a1}\,{x}^{2}}}-1/2\,\sqrt {2}\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) \left ( {\it a0}+1 \right ) \left ( {\frac {\sqrt {\pi }}{\sqrt {{\it a1}}}\sqrt {{\frac {{\it a1}}{\pi }}}}-1 \right ) \right ) {{\rm e}^{1/2\,{\it a1}\,{x}^{2}}}+ \left ( -x \left ( {\it a1}\,{x}^{2}+3\,{\it a0}+3 \right ) {\it b2}+{\it a1}\, \left ( {\it b1}\,{x}^{3}+3\,{\it b0}\,x-3\,z \right ) \right ) \sqrt {{\frac {{\it a1}}{\pi }}} \right ) {\frac {1}{\sqrt {{\frac {{\it a1}}{\pi }}}}}} \right ) {{\rm e}^{\int ^{x}\!1/6\,{\frac {1}{{\it a1}} \left ( -2\,{{\it c1} \left ( 3\,\sqrt {{\frac {{\it a1}}{\pi }}}{{\rm e}^{1/2\,{\it a1}\,{x}^{2}}}{\it b2}\, \left ( x+y \right ) {{\rm e}^{-1/2\,{\it a1}\,{x}^{2}}}-3/2\,{\it b2}\,\sqrt {2}\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}{\it \_a} \right ) \left ( {\it a0}+1 \right ) {{\rm e}^{1/2\,{\it a1}\,{{\it \_a}}^{2}}}-3/2\,{\it b2}\,\sqrt {2}\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) \left ( {\it a0}+1 \right ) \left ( {\frac {\sqrt {\pi }}{\sqrt {{\it a1}}}\sqrt {{\frac {{\it a1}}{\pi }}}}-1 \right ) {{\rm e}^{1/2\,{\it a1}\,{x}^{2}}}+ \left ( -x \left ( {\it a1}\,{x}^{2}+3\,{\it a0}+3 \right ) {\it b2}+{\it a1}\, \left ( {\it b1}\,{x}^{3}+3\,{\it b0}\,x-3\,z \right ) \right ) \sqrt {{\frac {{\it a1}}{\pi }}} \right ) {\frac {1}{\sqrt {{\frac {{\it a1}}{\pi }}}}}}-3\, \left ( \left ( -2\,x-2\,y \right ) {{\rm e}^{-1/2\,{\it a1}\,{x}^{2}}}+{\frac {\sqrt {2}\erf \left ( 1/2\,\sqrt {2}\sqrt {{\it a1}}x \right ) \sqrt {\pi } \left ( {\it a0}+1 \right ) }{\sqrt {{\it a1}}}} \right ) {\it b2}\,{\it c1}\,{{\rm e}^{1/2\,{\it a1}\,{{\it \_a}}^{2}}}-2\,{\it \_a}\,{\it c1}\, \left ( {\it a1}\,{{\it \_a}}^{2}+3\,{\it a0}+3 \right ) {\it b2}+6\, \left ( 1/3\,{\it b1}\,{\it c1}\,{{\it \_a}}^{3}+ \left ( {\it b0}\,{\it c1}+{\it c0}+{\it c2} \right ) {\it \_a}+s \right ) {\it a1} \right ) }{d{\it \_a}}}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x w_x + b y w_y + c z w_z = x(\lambda x+\beta y+\gamma z) w \]
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]== x*(lambda*x+beta*y+gama*z)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right ) e^{\frac {\beta x y}{a+b}+\frac {\text {gama} x z}{a+c}+\frac {\lambda x^2}{2 a}}\right \}\right \}\]
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)= x*(lambda*x+beta*y+gama*z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( y{x}^{-{\frac {b}{a}}},z{x}^{-{\frac {c}{a}}} \right ) {{\rm e}^{1/2\,{\frac { \left ( \left ( 2\,\beta \,y+2\,{\it gama}\,z+\lambda \,x \right ) {a}^{2}+ \left ( \lambda \, \left ( b+c \right ) x+2\,bz{\it gama}+2\,\beta \,cy \right ) a+xbc\lambda \right ) x}{a \left ( a+b \right ) \left ( a+c \right ) }}}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x^2 w_x + b x y w_y + c x z w_z = (\lambda x+\beta y+\gamma z) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^2*D[w[x, y,z], x] + b*x*y*D[w[x, y,z], y] +c*x*z*D[w[x,y,z],z]== (lambda*x+beta*y+gama*z)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to x^{\frac {\lambda }{a}} e^{-\frac {\frac {\beta y}{a-b}+\frac {\text {gama} z}{a-c}}{x}} c_1\left (y x^{-\frac {b}{a}},z x^{-\frac {c}{a}}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*x^2*diff(w(x,y,z),x)+b*x*y*diff(w(x,y,z),y)+c*x*z*diff(w(x,y,z),z)= (lambda*x+beta*y+gama*z)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={x}^{{\frac {\lambda }{a}}}{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}},z{x}^{-{\frac {c}{a}}} \right ) {{\rm e}^{{\frac {-y \left ( a-c \right ) \beta -z{\it gama}\, \left ( a-b \right ) }{ \left ( a-b \right ) \left ( a-c \right ) x}}}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x^2 w_x + b x y w_y + c z^2 w_z = k y^2 w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^2*D[w[x, y,z], x] + b*x*y*D[w[x, y,z], y] +c*z^2*D[w[x,y,z],z]== k*y^2*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{-\frac {k y^2}{a x-2 b x}} c_1\left (y x^{-\frac {b}{a}},\frac {c}{a x}-\frac {1}{z}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*x^2*diff(w(x,y,z),x)+b*x*y*diff(w(x,y,z),y)+c*z^2*diff(w(x,y,z),z)= k*y^2*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( y{x}^{-{\frac {b}{a}}},{\frac {ax-cz}{zxa}} \right ) {{\rm e}^{-{\frac {k{y}^{2}}{ \left ( a-2\,b \right ) x}}}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x^2 w_x + b y^2 w_y + c z^2 w_z = k x y w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^2*D[w[x, y,z], x] + b*y^2*D[w[x, y,z], y] +c*z^2*D[w[x,y,z],z]== k*x*y*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \left (\frac {a x}{y}\right )^{\frac {k x y}{a x-b y}} c_1\left (\frac {b}{a x}-\frac {1}{y},\frac {c}{a x}-\frac {1}{z}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*x^2*diff(w(x,y,z),x)+b*y^2*diff(w(x,y,z),y)+c*z^2*diff(w(x,y,z),z)= k*x*y*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ax-by}{axy}},{\frac {ax-cz}{zxa}} \right ) \left ( {\frac {ax}{y}} \right ) ^{{\frac {kxy}{ax-by}}}\]
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Added June 28, 2019.
Problem Chapter 8.2.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a x^2 w_x + b y^2 w_y + c z^2 w_z = (\lambda x^2+\beta y^2 + \gamma z^2) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^2*D[w[x, y,z], x] + b*y^2*D[w[x, y,z], y] +c*z^2*D[w[x,y,z],z]== (lambda*x^2+beta*y^2+gamma*z^2)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b}{a x}-\frac {1}{y},\frac {c}{a x}-\frac {1}{z}\right ) \exp \left (\frac {\beta y^2}{b y-a x}+\frac {\gamma z^2}{c z-a x}+\frac {\lambda x}{a}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*x^2*diff(w(x,y,z),x)+b*y^2*diff(w(x,y,z),y)+c*z^2*diff(w(x,y,z),z)= (lambda*x^2+beta*y^2+gamma*z^2)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ax-by}{axy}},{\frac {ax-cz}{zxa}} \right ) {{\rm e}^{1/8\,{\frac { \left ( -8\,\beta \,x{y}^{2}+8\,\lambda \,{x}^{3}-x{z}^{2} \right ) {a}^{2}+ \left ( 8\,c{y}^{2}z\beta + \left ( -8\,b\lambda \,{x}^{2}+b{z}^{2} \right ) y-8\,c{x}^{2}z\lambda \right ) a+8\,bcxyz\lambda }{a \left ( ax-by \right ) \left ( ax-cz \right ) }}}}\]
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