6.8.6 3.2
6.8.6.1 [1794] Problem 1
problem number 1794
Added July 2, 2019.
Problem Chapter 8.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a x^n w_y + b x^m w_z = (c e^{\lambda x} y + k e^{\beta x} z + s e^{\gamma x}) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*x^n*D[w[x, y,z], y] + b*x^m*D[w[x,y,z],z]== (c*Exp[lambda*x]*y+k*Exp[beta*x]*z+s*Exp[gamma*x])*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {-a x^{n+1}+n y+y}{n+1},\frac {-b x^{m+1}+m z+z}{m+1}\right ) \exp \left (-\frac {a c x^n (-\lambda x)^{-n} \Gamma (n+2,-\lambda x)}{\lambda ^2 (n+1)}+\frac {c e^{\lambda x} \left (-a x^{n+1}+n y+y\right )}{\lambda (n+1)}-\frac {b k x^m (-\beta x)^{-m} \Gamma (m+2,-\beta x)}{\beta ^2 (m+1)}+\frac {k e^{\beta x} \left (-b x^{m+1}+m z+z\right )}{\beta (m+1)}+\frac {s e^{\gamma x}}{\gamma }\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+a*x^n*diff(w(x,y,z),y)+c*x^m*diff(w(x,y,z),z)= (c*exp(lambda*x)*y+k*exp(beta*x)*z+s*exp(gamma*x))*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {-a \,x^{n +1}+y \left (n +1\right )}{n +1}, \frac {-x^{m +1} c +z \left (m +1\right )}{m +1}\right ) {\mathrm e}^{\frac {a \,\beta ^{2} \gamma \,x^{n} c \left (n +1\right ) \left (m +1\right ) \left (-n \Gamma \left (n , -\lambda x \right )+\Gamma \left (n +1\right )\right ) \left (-\lambda x \right )^{-n}+\gamma \,x^{m} \lambda ^{2} c k \left (n +1\right ) \left (m +1\right ) \left (-m \Gamma \left (m , -\beta x \right )+\Gamma \left (m +1\right )\right ) \left (-\beta x \right )^{-m}+\beta \gamma \,\lambda ^{2} c k \left (n +1\right ) x^{m +1}-\left (-x^{n +1} a \,\beta ^{2} c \gamma \lambda +\left (n +1\right ) \left (\beta ^{2} \gamma c \left (a \,x^{n}-\lambda y \right ) {\mathrm e}^{\lambda x}+\left (-\gamma \lambda k \left (\beta z -c \,x^{m}\right ) {\mathrm e}^{\beta x}+\left (-\beta \lambda s \,{\mathrm e}^{\gamma x}+\left (\beta c y +k \lambda z \right ) \gamma +\beta \lambda s \right ) \beta \right ) \lambda \right )\right ) \left (m +1\right )}{\left (n +1\right ) \left (m +1\right ) \beta ^{2} \lambda ^{2} \gamma }}\]
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6.8.6.2 [1795] Problem 2
problem number 1795
Added July 2, 2019.
Problem Chapter 8.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a e^{\lambda x} w_y + b x^m w_z = (c x^n y + k e^{\beta x} z + s e^{\gamma x}) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Exp[lambda*x]*D[w[x, y,z], y] + b*x^m*D[w[x,y,z],z]== (c*x^n*y+k*Exp[beta*x]*z+s*Exp[gamma*x])*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a e^{\lambda x}}{\lambda },\frac {-b x^{m+1}+m z+z}{m+1}\right ) \exp \left (\frac {a c x^n (-\lambda x)^{-n} \Gamma (n+1,-\lambda x)}{\lambda ^2}+\frac {c x^{n+1} \left (\lambda y-a e^{\lambda x}\right )}{\lambda (n+1)}-\frac {b k x^m (-\beta x)^{-m} \Gamma (m+2,-\beta x)}{\beta ^2 (m+1)}+\frac {k e^{\beta x} \left (-b x^{m+1}+m z+z\right )}{\beta (m+1)}+\frac {s e^{\gamma x}}{\gamma }\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+a*exp(lambda*x)*diff(w(x,y,z),y)+b*x^m*diff(w(x,y,z),z)= (c*x^n*y+k*exp(beta*x)*z+s*exp(gamma*x))*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {-a \,{\mathrm e}^{\lambda x}+y \lambda }{\lambda }, \frac {-b \,x^{m +1}+z \left (m +1\right )}{m +1}\right ) {\mathrm e}^{\frac {\int _{}^{x}\left (\textit {\_a}^{m +1} {\mathrm e}^{\beta \textit {\_a}} b k \lambda -{\mathrm e}^{\beta \textit {\_a}} x^{m +1} b k \lambda -\left (m +1\right ) \left (-{\mathrm e}^{\beta \textit {\_a}} k \lambda z -\textit {\_a}^{n} {\mathrm e}^{\lambda \textit {\_a}} a c -s \,{\mathrm e}^{\gamma \textit {\_a}} \lambda +\textit {\_a}^{n} \left (a \,{\mathrm e}^{\lambda x}-y \lambda \right ) c \right )\right )d \textit {\_a}}{\lambda \left (m +1\right )}}\]
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6.8.6.3 [1796] Problem 3
problem number 1796
Added July 2, 2019.
Problem Chapter 8.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a e^{\lambda x} w_y + b y w_z = (k e^{\beta x} z + s e^{\gamma x} ) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Exp[lambda*x]*D[w[x, y,z], y] + b*y*D[w[x,y,z],z]== (k*Exp[beta*x]*z+s*Exp[gamma*x])*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a e^{\lambda x}}{\lambda },\frac {a b e^{\lambda x} (\lambda x-1)}{\lambda ^2}-b x y+z\right ) \exp \left (\frac {a b k e^{x (\beta +\lambda )}}{\beta ^2 (\beta +\lambda )}+\frac {k e^{\beta x} (\beta z-b y)}{\beta ^2}+\frac {s e^{\gamma x}}{\gamma }\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+a*exp(lambda*x)*diff(w(x,y,z),y)+b*y*diff(w(x,y,z),z)= (k*exp(beta*x)*z+s*exp(gamma*x))*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {-a \,{\mathrm e}^{\lambda x}+y \lambda }{\lambda }, \frac {a b \left (\lambda x -1\right ) {\mathrm e}^{\lambda x}-\lambda ^{2} \left (b x y -z \right )}{\lambda ^{2}}\right ) {\mathrm e}^{\frac {a b \gamma k \,{\mathrm e}^{x \left (\beta +\lambda \right )}-\left (\beta +\lambda \right ) \left (k \gamma \left (b y -\beta z \right ) {\mathrm e}^{\beta x}-{\mathrm e}^{\gamma x} s \,\beta ^{2}\right )}{\gamma \,\beta ^{2} \left (\beta +\lambda \right )}}\]
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6.8.6.4 [1797] Problem 4
problem number 1797
Added July 2, 2019.
Problem Chapter 8.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a y^n w_y + b z^m w_z = (c e^{\lambda x} + k e^{\beta y}+ s e^{\gamma z} ) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*y^n*D[w[x, y,z], y] + b*z^m*D[w[x,y,z],z]== (c*Exp[lambda*x]+k*Exp[beta*y]+s*Exp[gamma*z])*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-a x-\frac {\left (\frac {1}{y}\right )^{n-1}}{n-1},-b x-\frac {\left (\frac {1}{z}\right )^{m-1}}{m-1}\right ) \exp \left (\frac {k \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {n}{n-1}} \left (-\beta \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {1}{1-n}}\right )^n \Gamma \left (1-n,-\beta \left (\left (\frac {1}{y}\right )^{n-1}\right )^{\frac {1}{1-n}}\right )}{a \beta }+\frac {s \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {m}{m-1}} \left (-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )^m \Gamma \left (1-m,-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )}{b \gamma }+\frac {c e^{\lambda x}}{\lambda }\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+a*y^n*diff(w(x,y,z),y)+b*z^m*diff(w(x,y,z),z)= (c*exp(lambda*x)+k*exp(beta*y)+s*exp(gamma*z))*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (y^{1-n}+\left (n -1\right ) x a , z^{1-m}+\left (m -1\right ) x b \right ) {\mathrm e}^{\int _{}^{x}\left (c \,{\mathrm e}^{\lambda \textit {\_a}}+k \,{\mathrm e}^{\beta \left (y^{1-n}+\left (\left (1-n \right ) \textit {\_a} +x n -x \right ) a \right )^{-\frac {1}{n -1}}}+s \,{\mathrm e}^{\gamma \left (z^{1-m}+\left (\left (1-m \right ) \textit {\_a} +x m -x \right ) b \right )^{-\frac {1}{m -1}}}\right )d \textit {\_a}}\]
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6.8.6.5 [1798] Problem 5
problem number 1798
Added July 2, 2019.
Problem Chapter 8.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a e^{\beta y} w_y + b z^m w_z = (c e^{\lambda x} + k y^n+ s e^{\gamma z} ) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Exp[beta*y]*D[w[x, y,z], y] + b*z^m*D[w[x,y,z],z]== (c*Exp[lambda*x]+k*y^n+s*Exp[gamma*z])*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {a \beta x+e^{-\beta y}}{\beta },-b x-\frac {\left (\frac {1}{z}\right )^{m-1}}{m-1}\right ) \exp \left (-\frac {k \left (-\log \left (e^{-\beta y}\right )\right )^{-n} \left (-\frac {\log \left (e^{-\beta y}\right )}{\beta }\right )^n \Gamma \left (n+1,-\log \left (e^{-\beta y}\right )\right )}{a \beta }+\frac {s \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {m}{m-1}} \left (-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )^m \Gamma \left (1-m,-\gamma \left (\left (\frac {1}{z}\right )^{m-1}\right )^{\frac {1}{1-m}}\right )}{b \gamma }+\frac {c e^{\lambda x}}{\lambda }\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+a*exp(beta*y)*diff(w(x,y,z),y)+b*z^m*diff(w(x,y,z),z)= (c*exp(lambda*x)+k*y^n+s*exp(gamma*z))*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {-{\mathrm e}^{-\beta y}-a \beta x}{a \beta }, z^{1-m}+x \left (m -1\right ) b \right ) {\mathrm e}^{\int _{}^{x}\left (c \,{\mathrm e}^{\lambda \textit {\_a}}+k {\left (\frac {\ln \left (\frac {1}{{\mathrm e}^{-\beta y}+\left (x -\textit {\_a} \right ) a \beta }\right )}{\beta }\right )}^{n}+s \,{\mathrm e}^{\gamma \left (z^{1-m}+\left (\left (1-m \right ) \textit {\_a} +x m -x \right ) b \right )^{-\frac {1}{m -1}}}\right )d \textit {\_a}}\]
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6.8.6.6 [1799] Problem 6
problem number 1799
Added July 2, 2019.
Problem Chapter 8.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (y^2+ b y+ a e^{\alpha y}(y-b)-b^2) w_y + (z^2+c(x z-1)e^{\beta x}) w_z = k e^{\lambda x} w \]
Mathematica ✗
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (y^2+ b*y+ a*Exp[alpha*y]*(y-b)-b^2)*D[w[x, y,z], y] + (z^2+c*(x*z-1)*Exp[beta*x])*D[w[x,y,z],z]== k*Exp[lambda*x]*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+(y^2+ b*y+ a*exp(alpha*y)*(y-b)-b^2)*diff(w(x,y,z),y)+(z^2+c*(x*z-1)*exp(beta*x))*diff(w(x,y,z),z)= k*exp(lambda*x)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
\[w \left (x , y , z\right )=c_{1} {\mathrm e}^{\textit {\_c}_{2} \int \frac {1}{a \left (b -y \right ) {\mathrm e}^{\alpha y}+b^{2}-b y -y^{2}}d y} f_{10} \left (x , z\right )\boldsymbol {\operatorname {where}}\left [\left \{\left (z^{2}+c \left (x z -1\right ) {\mathrm e}^{\beta x}\right ) \left (\frac {\partial }{\partial z}f_{10} \left (x , z\right )\right )-k \,{\mathrm e}^{\lambda x} f_{10} \left (x , z\right )-\textit {\_c}_{2} f_{10} \left (x , z\right )+\frac {\partial }{\partial x}f_{10} \left (x , z\right )=0\right \}\right ]\]
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6.8.6.7 [1800] Problem 7
problem number 1800
Added July 2, 2019.
Problem Chapter 8.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (y^2+ a e^{\alpha x}(x+1)) w_y + (c e^{\beta x} z^2 + b e^{-\beta x}) w_z = k e^{\lambda x} w \]
Mathematica ✗
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (y^2+ a*Exp[alpha*x]*(x+1))*D[w[x, y,z], y] + (c*Exp[beta*x]*z^2+b*Exp[-beta*x])*D[w[x,y,z],z]== k*Exp[lambda*x]*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✗
restart;
local gamma;
pde := diff(w(x,y,z),x)+(y^2+ a*exp(alpha*x)*(x+1))*diff(w(x,y,z),y)+(c*exp(beta*x)*z^2+b*exp(-beta*x))*diff(w(x,y,z),z)= k*exp(lambda*x)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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6.8.6.8 [1801] Problem 8
problem number 1801
Added July 2, 2019.
Problem Chapter 8.3.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a e^{\alpha x}y^2+b e^{-\alpha x}) w_y + (d e^{\beta x} z^2 + c e^{\gamma x}(\gamma -c d e^{(\beta +\gamma )x})) w_z = k e^{\lambda x} w \]
Mathematica ✗
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (a*Exp[alpha*x]*y^2+b*Exp[-alpha*x])*D[w[x, y,z], y] + (d*Exp[beta*x]*z^2 + c*Exp[gamma*x]*(gamma-c*d*Exp[(beta+gamma)*x]))*D[w[x,y,z],z]== k*Exp[lambda*x]*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+(a*exp(alpha*x)*y^2+b*exp(-alpha*x))*diff(w(x,y,z),y)+(d*exp(beta*x)*z^2 + c*exp(gamma*x)*(gamma-c*d*exp((beta+gamma)*x)))*diff(w(x,y,z),z)= k*exp(lambda*x)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[\text {Expression too large to display}\]
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6.8.6.9 [1802] Problem 9
problem number 1802
Added July 2, 2019.
Problem Chapter 8.3.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 e^{\lambda _1 x} y+b_1 e^{\beta _1 x} y^k) w_y + (a_2 e^{\lambda _2 x} z+b_2 e^{\beta _2 x} z^m) w_z = c x^s w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (a1*Exp[lambda1*x]*y+b1*Exp[beta1*x]*y^k)*D[w[x, y,z], y] + (a2*Exp[lambda2*x]*z+b2*Exp[beta2*x]*z^m)*D[w[x,y,z],z]== c*x^s*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^{s+1}}{s+1}} c_1\left ((k-1) \int _1^x\text {b1} e^{\frac {\text {a1} e^{\text {lambda1} K[1]} (k-1)}{\text {lambda1}}+\text {beta1} K[1]}dK[1]+y^{1-k} e^{\frac {\text {a1} (k-1) e^{\text {lambda1} x}}{\text {lambda1}}},(m-1) \int _1^x\text {b2} e^{\frac {\text {a2} e^{\text {lambda2} K[2]} (m-1)}{\text {lambda2}}+\text {beta2} K[2]}dK[2]+z^{1-m} e^{\frac {\text {a2} (m-1) e^{\text {lambda2} x}}{\text {lambda2}}}\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+(a1*exp(lambda1*x)*y+b1*exp(beta1*x)*y^k)*diff(w(x,y,z),y)+(a2*exp(lambda2*x)*z+b2*exp(beta2*x)*z^m)*diff(w(x,y,z),z)= c*x^s*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\operatorname {b1} \left (k -1\right ) \int {\mathrm e}^{\frac {\operatorname {a1} \,{\mathrm e}^{\lambda \operatorname {1} x} \left (k -1\right )+\beta \operatorname {1} x \lambda \operatorname {1} }{\lambda \operatorname {1}}}d x +y^{1-k} {\mathrm e}^{\frac {\operatorname {a1} \,{\mathrm e}^{\lambda \operatorname {1} x} \left (k -1\right )}{\lambda \operatorname {1}}}, \operatorname {b2} \left (m -1\right ) \int {\mathrm e}^{\frac {\operatorname {a2} \,{\mathrm e}^{\lambda \operatorname {2} x} \left (m -1\right )+\beta \operatorname {2} x \lambda \operatorname {2} }{\lambda \operatorname {2}}}d x +z^{1-m} {\mathrm e}^{\frac {\operatorname {a2} \,{\mathrm e}^{\lambda \operatorname {2} x} \left (m -1\right )}{\lambda \operatorname {2}}}\right ) {\mathrm e}^{\frac {c \,x^{s +1}}{s +1}}\]
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6.8.6.10 [1803] Problem 10
problem number 1803
Added July 2, 2019.
Problem Chapter 8.3.2.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 e^{\beta _1 x} y+b_1 e^{\gamma _1 x} y^k) w_y + (a_2 e^{\beta _2 x}+b_2 e^{\gamma _2 x+\lambda _2 z} ) w_z = c x^s w \]
Mathematica ✗
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (a1*Exp[beta1*x]*y+b1*Exp[gamma1*x]*y^k)*D[w[x, y,z], y] + (a2*Exp[beta2*x]+b2*Exp[gamma2*x+lambda2*z])*D[w[x,y,z],z]== c*x^s*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+(a1*exp(beta1*x)*y+b1*exp(gamma1*x)*y^k)*diff(w(x,y,z),y)+ (a2*exp(beta2*x)+b2*exp(gamma2*x+lambda2*z))*diff(w(x,y,z),z)= c*x^s*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\operatorname {b1} \left (k -1\right ) \int {\mathrm e}^{\frac {\operatorname {a1} \,{\mathrm e}^{\beta \operatorname {1} x} \left (k -1\right )+\gamma \operatorname {1} x \beta \operatorname {1} }{\beta \operatorname {1}}}d x +y^{1-k} {\mathrm e}^{\frac {\operatorname {a1} \,{\mathrm e}^{\beta \operatorname {1} x} \left (k -1\right )}{\beta \operatorname {1}}}, \frac {-{\mathrm e}^{\frac {\lambda \operatorname {2} \left (\operatorname {a2} \,{\mathrm e}^{\beta \operatorname {2} x}-\beta \operatorname {2} z \right )}{\beta \operatorname {2}}}-\operatorname {b2} \int {\mathrm e}^{\frac {\lambda \operatorname {2} \operatorname {a2} \,{\mathrm e}^{\beta \operatorname {2} x}+\gamma \operatorname {2} x \beta \operatorname {2} }{\beta \operatorname {2}}}d x \lambda \operatorname {2} }{\lambda \operatorname {2}}\right ) {\mathrm e}^{\frac {c \,x^{s +1}}{s +1}}\]
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6.8.6.11 [1804] Problem 11
problem number 1804
Added July 2, 2019.
Problem Chapter 8.3.2.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a_1 x^n+b_1 x^m e^{\lambda y} ) w_y + (a_2 x^k+ b_2 x^r e^{\beta z}) w_z = c x^s w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (a1*x^n+b1*x^m*Exp[lambda*y] )*D[w[x, y,z], y] + (a2*x^k+b2*x^r*Exp[beta*z])*D[w[x,y,z],z]== c*x^s*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {c x^{s+1}}{s+1}} c_1\left (\frac {(n+1) e^{-\frac {\lambda \left (-\text {a1} x^{n+1}+n y+y\right )}{n+1}}-\text {b1} \lambda x^{m+1} \left (-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )^{-\frac {m+1}{n+1}} \Gamma \left (\frac {m+1}{n+1},-\frac {\text {a1} \lambda x^{n+1}}{n+1}\right )}{\text {a1} \text {b1} \lambda ^2 (n+1) (m-n)},\frac {\text {b2} \beta x^{r+1} \left (-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )^{-\frac {r+1}{k+1}} \Gamma \left (\frac {r+1}{k+1},-\frac {\text {a2} \beta x^{k+1}}{k+1}\right )-(k+1) e^{-\frac {\beta \left (-\text {a2} x^{k+1}+k z+z\right )}{k+1}}}{\text {a2} \text {b2} \beta ^2 (k+1) (k-r)}\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+(a1*x^n+b1*x^m*exp(lambda*y) )*diff(w(x,y,z),y)+ (a2*x^k+b2*x^r*exp(beta*z))*diff(w(x,y,z),z)= c*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 ) = f_{1} \left (-\frac {x^{-n} \left (-\frac {x \lambda \operatorname {a1} \,x^{n}}{n +1}\right )^{\frac {-m -n -2}{2 n +2}} \left (-{\mathrm e}^{\frac {x \lambda \operatorname {a1} \,x^{n}}{2 n +2}} \operatorname {b1} \,x^{m} \left (n +1\right )^{2} \left (-x \lambda \operatorname {a1} \,x^{n}+m +n +2\right ) \operatorname {WhittakerM}\left (\frac {m -n}{2 n +2}, \frac {m +2 n +3}{2 n +2}, -\frac {x \lambda \operatorname {a1} \,x^{n}}{n +1}\right )+\left (-{\mathrm e}^{\frac {x \lambda \operatorname {a1} \,x^{n}}{2 n +2}} \operatorname {b1} \,x^{m} \left (n +1\right ) \left (m +n +2\right ) \operatorname {WhittakerM}\left (\frac {m +n +2}{2 n +2}, \frac {m +2 n +3}{2 n +2}, -\frac {x \lambda \operatorname {a1} \,x^{n}}{n +1}\right )+{\mathrm e}^{-\frac {\left (-\operatorname {a1} x \,x^{n}+y \left (n +1\right )\right ) \lambda }{n +1}} \operatorname {a1} \,x^{n} \left (-\frac {x \lambda \operatorname {a1} \,x^{n}}{n +1}\right )^{\frac {m +n +2}{2 n +2}} \left (m +1\right ) \left (m +2 n +3\right )\right ) \left (m +n +2\right )\right )}{\operatorname {a1} \lambda \left (m +1\right ) \left (m +2 n +3\right ) \left (m +n +2\right )}, \frac {\left ({\mathrm e}^{\frac {x \beta \operatorname {a2} \,x^{k}}{2 k +2}} \operatorname {b2} \,x^{r} \left (k +1\right )^{2} \left (-x \beta \operatorname {a2} \,x^{k}+k +r +2\right ) \operatorname {WhittakerM}\left (\frac {-k +r}{2 k +2}, \frac {2 k +r +3}{2 k +2}, -\frac {x \beta \operatorname {a2} \,x^{k}}{k +1}\right )+\left (k +r +2\right ) \left ({\mathrm e}^{\frac {x \beta \operatorname {a2} \,x^{k}}{2 k +2}} \operatorname {b2} \,x^{r} \left (k +1\right ) \left (k +r +2\right ) \operatorname {WhittakerM}\left (\frac {k +r +2}{2 k +2}, \frac {2 k +r +3}{2 k +2}, -\frac {x \beta \operatorname {a2} \,x^{k}}{k +1}\right )-2 \left (r +1\right ) \left (-\frac {x \beta \operatorname {a2} \,x^{k}}{k +1}\right )^{\frac {k +r +2}{2 k +2}} \operatorname {a2} \left (k +\frac {r}{2}+\frac {3}{2}\right ) {\mathrm e}^{-\frac {\left (-\operatorname {a2} x \,x^{k}+z \left (k +1\right )\right ) \beta }{k +1}} x^{k}\right )\right ) x^{-k} \left (-\frac {x \beta \operatorname {a2} \,x^{k}}{k +1}\right )^{\frac {-k -r -2}{2 k +2}}}{\operatorname {a2} \beta \left (r +1\right ) \left (k +r +2\right ) \left (2 k +r +3\right )}\right ) {\mathrm e}^{\frac {c \,x^{s} x}{s +1}}\]
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