6.6.25 8.3
6.6.25.1 [1567] Problem 1
problem number 1567
Added May 31, 2019.
Problem Chapter 6.8.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + f(x,y) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +f[x,y]*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\int _1^x\frac {f\left (K[1],y+\frac {b (K[1]-x)}{a}\right )}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+f(x,y)*diff(w(x,y,z),z)= 0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -x b}{a}, -\frac {\int _{}^{x}f \left (\textit {\_a} , \frac {a y -b \left (x -\textit {\_a} \right )}{a}\right )d \textit {\_a}}{a}+z \right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.6.25.2 [1568] Problem 2
problem number 1568
Added May 31, 2019.
Problem Chapter 6.8.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + f(x,y) g(z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +f[x,y]*g[z]*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\int _1^z\frac {1}{g(K[1])}dK[1]-\int _1^x\frac {f\left (K[2],y+\frac {b (K[2]-x)}{a}\right )}{a}dK[2]\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+f(x,y)*g(z)*diff(w(x,y,z),z)= 0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -b x}{a}, -\int _{}^{x}f \left (\textit {\_a} , \frac {a y -b \left (x -\textit {\_a} \right )}{a}\right )d \textit {\_a} +a \int \frac {1}{g \left (z \right )}d z \right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.6.25.3 [1569] Problem 3
problem number 1569
Added May 31, 2019.
Problem Chapter 6.8.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ x w_x + y w_y + (z+f(x,y)) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = x*D[w[x, y,z], x] + y*D[w[x, y,z], y] +(z+f[x,y])*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {z}{x}-\int _1^x\frac {f\left (K[1],\frac {y K[1]}{x}\right )}{K[1]^2}dK[1]\right )\right \}\right \}\]
Maple ✓
restart;
pde := x*diff(w(x,y,z),x)+ y*diff(w(x,y,z),y)+(z+f(x,y))*diff(w(x,y,z),z)= 0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {y}{x}, \frac {-\int _{}^{x}\frac {f \left (\textit {\_a} , \frac {y \textit {\_a}}{x}\right )}{\textit {\_a}^{2}}d \textit {\_a} x +z}{x}\right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.6.25.4 [1570] Problem 4
problem number 1570
Added May 31, 2019.
Problem Chapter 6.8.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x w_x + b y w_y + f(x,y) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*x*D[w[x, y,z], x] + b*y*D[w[x, y,z], y] +f[x,y]*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y x^{-\frac {b}{a}},z-\int _1^x\frac {f\left (K[1],x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )}{a K[1]}dK[1]\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*x*diff(w(x,y,z),x)+ b*y*diff(w(x,y,z),y)+f(x,y)*diff(w(x,y,z),z)= 0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (y \,x^{-\frac {b}{a}}, -\frac {\int _{}^{x}\frac {f \left (\textit {\_a} , y \,x^{-\frac {b}{a}} \textit {\_a}^{\frac {b}{a}}\right )}{\textit {\_a}}d \textit {\_a}}{a}+z \right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.6.25.5 [1571] Problem 5
problem number 1571
Added May 31, 2019.
Problem Chapter 6.8.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x w_x + b y w_y + f(x,y) g(z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*x*D[w[x, y,z], x] + b*y*D[w[x, y,z], y] +f[x,y]*g[x]*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y x^{-\frac {b}{a}},z-\int _1^x\frac {f\left (K[1],x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right ) g(K[1])}{a K[1]}dK[1]\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*x*diff(w(x,y,z),x)+ b*y*diff(w(x,y,z),y)+f(x,y)*g(z)*diff(w(x,y,z),z)= 0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (y \,x^{-\frac {b}{a}}, -\int _{}^{x}\frac {f \left (\textit {\_a} , y \,x^{-\frac {b}{a}} \textit {\_a}^{\frac {b}{a}}\right )}{\textit {\_a}}d \textit {\_a} +a \int \frac {1}{g \left (z \right )}d z \right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.6.25.6 [1572] Problem 6
problem number 1572
Added May 31, 2019.
Problem Chapter 6.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x)) w_y + (g(x,y) z + h(x,y) )w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (f1[x]*y+f2[x])*D[w[x, y,z], y] +(g[x,y]*z+h[x,y])*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2],z \exp \left (-\int _1^xg\left (K[3],\exp \left (\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[3]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )dK[3]\right )-\int _1^x\exp \left (-\int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [\log \left (\exp \left (\int _1^xg\left (K[3],\exp \left (\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[3]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )dK[3]\right )\right ),\{K[3],1,x\}\right ]dK[3]\right ) h\left (K[4],\exp \left (\int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[4]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )dK[4]\right )\right \}\right \}\]
Maple ✓
restart;
pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x))*diff(w(x,y,z),y)+(g(x,y)*z+h(x,y))*diff(w(x,y,z),z)= 0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left ({\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x} y -\int \operatorname {f2} \left (x \right ) {\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x}d x , -\int _{}^{x}h \left (\textit {\_f} , \left (\int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{-\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} +{\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x} y -\int \operatorname {f2} \left (x \right ) {\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x}d x \right ) {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}\right ) {\mathrm e}^{-\int g \left (\textit {\_f} , \left (\int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{-\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} +{\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x} y -\int \operatorname {f2} \left (x \right ) {\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x}d x \right ) {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}\right )d \textit {\_f}}d \textit {\_f} +{\mathrm e}^{-\int _{}^{x}g \left (\textit {\_f} , \left (\int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{-\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} +{\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x} y -\int \operatorname {f2} \left (x \right ) {\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x}d x \right ) {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}\right )d \textit {\_f}} z \right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.6.25.7 [1573] Problem 7
problem number 1573
Added May 31, 2019.
Problem Chapter 6.8.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x) y^k) w_y + (g(x,y) z + h(x,y) z^m )w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (f1[x]*y+f2[x]*y^k)*D[w[x, y,z], y] +(g[x,y]*z+h[x,y]*z^m)*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left ((k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\text {f1}(K[1])dK[1]\right ),(m-1) \int _1^x\exp \left ((m-1) \int _1^{K[4]}\text {InverseFunction}[\text {Inactive}[\text {Integrate}],1,2]\left [-\frac {\log \left (\exp \left (-\left ((m-1) \int _1^xg\left (K[3],\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[3]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )dK[3]\right )\right )\right )}{m-1},\{K[3],1,x\}\right ]dK[3]\right ) h\left (K[4],\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[4]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )dK[4]+z^{1-m} \exp \left ((m-1) \int _1^xg\left (K[3],\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[3]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )dK[3]\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+(g(x,y)*z+h(x,y)*z^m)*diff(w(x,y,z),z)= 0;
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 (\left (k -1\right ) \int \operatorname {f2} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}, \left (m -1\right ) \int _{}^{x}h \left (\textit {\_f} , {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}} \left (\left (1-k \right ) \int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f} \left (k -1\right )}d \textit {\_f} +\left (k -1\right ) \int \operatorname {f2} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}\right )^{-\frac {1}{k -1}}\right ) {\mathrm e}^{\int g \left (\textit {\_f} , {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}} \left (\left (1-k \right ) \int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f} \left (k -1\right )}d \textit {\_f} +\left (k -1\right ) \int \operatorname {f2} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}\right )^{-\frac {1}{k -1}}\right )d \textit {\_f} \left (m -1\right )}d \textit {\_f} +z^{1-m} {\mathrm e}^{\int _{}^{x}g \left (\textit {\_f} , {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}} \left (\left (1-k \right ) \int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f} \left (k -1\right )}d \textit {\_f} +\left (k -1\right ) \int \operatorname {f2} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}\right )^{-\frac {1}{k -1}}\right )d \textit {\_f} \left (m -1\right )}\right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.6.25.8 [1574] Problem 8
problem number 1574
Added May 31, 2019.
Problem Chapter 6.8.3.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x) y^k) w_y + (g(x,y) + h(x,y) e^{\lambda z} )w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (f1[x]*y+f2[x]*y^k)*D[w[x, y,z], y] +(g[x,y]*z+h[x,y]*Exp[lambda*z])*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart;
pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+(g(x,y)*z+h(x,y)*exp(lambda*z))*diff(w(x,y,z),z)= 0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.6.25.9 [1575] Problem 9
problem number 1575
Added May 31, 2019.
Problem Chapter 6.8.3.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) +f_2(x) e^{\lambda y}) w_y + (g(x,y) z + h(x,y) z^k )w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (f1[x]+f2[x]*Exp[lambda*y])*D[w[x, y,z], y] +(g[x,y]*z+h[x,y]*z^k)*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart;
pde := diff(w(x,y,z),x)+ (f1(x)+f2(x)*exp(lambda*y))*diff(w(x,y,z),y)+(g(x,y)*z+h(x,y)*z^k)*diff(w(x,y,z),z)= 0;
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}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}-\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda }{\lambda }, \left (k -1\right ) \int _{}^{x}h \left (\textit {\_h} , \frac {\ln \left (\frac {1}{\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda -\int \operatorname {f2} \left (\textit {\_g} \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (\textit {\_g} \right )d \textit {\_g}}d \textit {\_g} \lambda +{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}}\right )+\lambda \int \operatorname {f1} \left (\textit {\_h} \right )d \textit {\_h}}{\lambda }\right ) {\mathrm e}^{\int g \left (\textit {\_h} , \frac {\ln \left (\frac {1}{-\int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} \lambda +\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda +{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}}\right )+\lambda \int \operatorname {f1} \left (\textit {\_h} \right )d \textit {\_h}}{\lambda }\right )d \textit {\_h} \left (k -1\right )}d \textit {\_h} +z^{1-k} {\mathrm e}^{\int _{}^{x}g \left (\textit {\_h} , \frac {\ln \left (\frac {1}{-\int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} \lambda +\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda +{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}}\right )+\lambda \int \operatorname {f1} \left (\textit {\_h} \right )d \textit {\_h}}{\lambda }\right )d \textit {\_h} \left (k -1\right )}\right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.6.25.10 [1576] Problem 10
problem number 1576
Added May 31, 2019.
Problem Chapter 6.8.3.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) +f_2(x) e^{\lambda y}) w_y + (g(x,y) + h(x,y) e^{\beta z} )w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (f1[x]+f2[x]*Exp[lambda*y])*D[w[x, y,z], y] +(g[x,y]+h[x,y]*Exp[beta*z])*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart;
pde := diff(w(x,y,z),x)+ (f1(x)+f2(x)*exp(lambda*y))*diff(w(x,y,z),y)+(g(x,y)+h(x,y)*exp(beta*z))*diff(w(x,y,z),z)= 0;
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 {-\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda -{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}}{\lambda }, \frac {-\int _{}^{x}h \left (\textit {\_h} , \frac {\ln \left (\frac {1}{-\int \operatorname {f2} \left (\textit {\_g} \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (\textit {\_g} \right )d \textit {\_g}}d \textit {\_g} \lambda +\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda +{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}}\right )+\lambda \int \operatorname {f1} \left (\textit {\_h} \right )d \textit {\_h}}{\lambda }\right ) {\mathrm e}^{\beta \int g \left (\textit {\_h} , \frac {\ln \left (\frac {1}{-\int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} \lambda +\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda +{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}}\right )+\lambda \int \operatorname {f1} \left (\textit {\_h} \right )d \textit {\_h}}{\lambda }\right )d \textit {\_h}}d \textit {\_h} \beta -{\mathrm e}^{\beta \left (-z +\int _{}^{x}g \left (\textit {\_h} , \frac {\ln \left (\frac {1}{-\int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} \lambda +\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda +{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}}\right )+\lambda \int \operatorname {f1} \left (\textit {\_h} \right )d \textit {\_h}}{\lambda }\right )d \textit {\_h} \right )}}{\beta }\right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.6.25.11 [1577] Problem 11
problem number 1577
Added May 31, 2019.
Problem Chapter 6.8.3.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f_1(x) g_1(y) w_x + f_2(x) g_2(y) w_y + (h_1(x,y) + h_2(x,y) z^m )w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"];
pde = f1[x]*g1[y]*D[w[x, y,z], x] + f2[x]*g2[y]*D[w[x, y,z], y] +(h1[x,y]+h2[x,y]*z^m)*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart;
pde := f1(x)*g1(y)*diff(w(x,y,z),x)+ f2(x)*g2(y)*diff(w(x,y,z),y)+(h1(x,y)+h2(x,y)*z^m)*diff(w(x,y,z),z)= 0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.6.25.12 [1578] Problem 12
problem number 1578
Added May 31, 2019.
Problem Chapter 6.8.3.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f_1(x) g_1(y) w_x + f_2(x) g_2(y) w_y + (h_1(x,y) + h_2(x,y) e^{\lambda z} )w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"];
pde = f1[x]*g1[y]*D[w[x, y,z], x] + f2[x]*g2[y]*D[w[x, y,z], y] +(h1[x,y]+h2[x,y]*Exp[lambda*z])*D[w[x,y,z],z]==0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart;
pde := f1(x)*g1(y)*diff(w(x,y,z),x)+ f2(x)*g2(y)*diff(w(x,y,z),y)+(h1(x,y)+h2(x,y)*exp(lambda*z))*diff(w(x,y,z),z)= 0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()