6.7.25 8.2
6.7.25.1 [1731] Problem 1
problem number 1731
Added June 27, 2019.
Problem Chapter 7.8.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f(x) w_x + g(y) w_y + h(z) w_z = \Phi (x) + \Psi (x) + \chi (x) \]
Mathematica ✗
ClearAll["Global`*"];
pde = f[x]*D[w[x, y,z], x] + g[y]*D[w[x, y,z], y] + h[z]*D[w[x,y,z],z]== phi[x]+psi[x]+chi[x];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := f(x)*diff(w(x,y,z),x)+ g(y)*diff(w(x,y,z),y)+ h(z)*diff(w(x,y,z),z)= phi(x)+psi(x)+chi(x);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int \frac {\phi \left (x \right )+\psi \left (x \right )+\chi \left (x \right )}{f \left (x \right )}d x +f_{1} \left (-\int \frac {1}{f \left (x \right )}d x +\int \frac {1}{g \left (y \right )}d y , -\int \frac {1}{f \left (x \right )}d x +\int \frac {1}{h \left (z \right )}d z \right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.7.25.2 [1732] Problem 2
problem number 1732
Added June 27, 2019.
Problem Chapter 7.8.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f(x) w_x + z w_y + g(y) w_z = h_2(x)+h_1(y) \]
Mathematica ✗
ClearAll["Global`*"];
pde = f[x]*D[w[x, y,z], x] + z*D[w[x, y,z], y] + g[y]*D[w[x,y,z],z]== h2[x]+h1[y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := f(x)*diff(w(x,y,z),x)+ z*diff(w(x,y,z),y)+ g(y)*diff(w(x,y,z),z)= h2(x)+h1(y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{y}\frac {\operatorname {h2} \left (\operatorname {RootOf}\left (\int \frac {1}{\sqrt {2 \int g \left (\textit {\_b} \right )d \textit {\_b} +z^{2}-2 \int g \left (y \right )d y}}d \textit {\_b} -\int _{}^{\textit {\_Z}}\frac {1}{f \left (\textit {\_b} \right )}d \textit {\_b} -\int _{}^{y}\frac {1}{\sqrt {2 \int g \left (\textit {\_b} \right )d \textit {\_b} +z^{2}-2 \int g \left (y \right )d y}}d \textit {\_b} +\int \frac {1}{f \left (x \right )}d x \right )\right )+\operatorname {h1} \left (\textit {\_b} \right )}{\sqrt {2 \int g \left (\textit {\_b} \right )d \textit {\_b} +z^{2}-2 \int g \left (y \right )d y}}d \textit {\_b} +f_{1} \left (z^{2}-2 \int g \left (y \right )d y , -\int _{}^{y}\frac {1}{\sqrt {2 \int g \left (\textit {\_b} \right )d \textit {\_b} +z^{2}-2 \int g \left (y \right )d y}}d \textit {\_b} +\int \frac {1}{f \left (x \right )}d x \right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.7.25.3 [1733] Problem 3
problem number 1733
Added June 27, 2019.
Problem Chapter 7.8.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ f_1(x) w_x + f_2(x) g(y) w_y + f_3(x) h(z) w_z = f_4(x) \]
Mathematica ✗
ClearAll["Global`*"];
pde = f1[x]*D[w[x, y,z], x] + f2[x]*g[y]*D[w[x, y,z], y] + f3[x]*h[z]*D[w[x,y,z],z]== f4[x];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := f1(x)*diff(w(x,y,z),x)+ f2(x)*g(y)*diff(w(x,y,z),y)+ f3(x)*h(z)*diff(w(x,y,z),z)= f4(x);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int \frac {\operatorname {f4} \left (x \right )}{\operatorname {f1} \left (x \right )}d x +f_{1} \left (-\int \frac {\operatorname {f2} \left (x \right )}{\operatorname {f1} \left (x \right )}d x +\int \frac {1}{g \left (y \right )}d y , -\int \frac {\operatorname {f3} \left (x \right )}{\operatorname {f1} \left (x \right )}d x +\int \frac {1}{h \left (z \right )}d z \right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.7.25.4 [1734] Problem 4
problem number 1734
Added June 27, 2019.
Problem Chapter 7.8.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x)) w_y + (g_1(x) z+g_2(y)) w_z = h_1(x)+h_2(y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (f1[x]*y+f2[x])*D[w[x, y,z], y] + (g1[x]*z+g2[y])*D[w[x,y,z],z]== h1[x]+h2[y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (\text {h1}(K[5])+\text {h2}\left (\exp \left (\int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[5]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )\right )dK[5]+c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2],z \exp \left (-\int _1^x\text {g1}(K[3])dK[3]\right )-\int _1^x\exp \left (-\int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}\left (\exp \left (\int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[4]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )\right )dK[4]\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x))*diff(w(x,y,z),y)+ (g1(x)*z+g2(y))*diff(w(x,y,z),z)= h1(x)+h2(y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\left (\operatorname {h1} \left (\textit {\_f} \right )+\operatorname {h2} \left (\left (\int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{-\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} +{\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x} y -\int \operatorname {f2} \left (x \right ) {\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x}d x \right ) {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}\right )\right )d \textit {\_f} +f_{1} \left ({\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x} y -\int \operatorname {f2} \left (x \right ) {\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x}d x , {\mathrm e}^{-\int \operatorname {g1} \left (x \right )d x} z -\int _{}^{x}\operatorname {g2} \left (\left (\int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{-\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} +{\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x} y -\int \operatorname {f2} \left (x \right ) {\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x}d x \right ) {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}\right ) {\mathrm e}^{-\int \operatorname {g1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} \right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.7.25.5 [1735] Problem 5
problem number 1735
Added June 27, 2019.
Problem Chapter 7.8.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x) y^k) w_y + (g_1(x) z+g_2(y) z^m) w_z = h_1(x)+h_2(y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (f1[x]*y+f2[x]*y^k)*D[w[x, y,z], y] + (g1[x]*z+g2[y]*z^m)*D[w[x,y,z],z]== h1[x]+h2[y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (\text {h1}(K[5])+\text {h2}\left (\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[5]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )\right )dK[5]+c_1\left ((k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\text {f1}(K[1])dK[1]\right ),(m-1) \int _1^x\exp \left ((m-1) \int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}\left (\left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]-(k-1) \int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) (k-1) \int _1^{K[4]}\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2] y^k+\exp \left (k \int _1^x\text {f1}(K[1])dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )dK[4]+z^{1-m} \exp \left ((m-1) \int _1^x\text {g1}(K[3])dK[3]\right )\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+ (g1(x)*z+g2(y)*z^m)*diff(w(x,y,z),z)= h1(x)+h2(y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\left (\operatorname {h1} \left (\textit {\_f} \right )+\operatorname {h2} \left (\left (\left (1-k \right ) \int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f} \left (k -1\right )}d \textit {\_f} +\left (k -1\right ) \int \operatorname {f2} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}\right )\right )d \textit {\_f} +f_{1} \left (\left (k -1\right ) \int \operatorname {f2} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}, \left (m -1\right ) \int _{}^{x}{\mathrm e}^{\int \operatorname {g1} \left (\textit {\_f} \right )d \textit {\_f} \left (m -1\right )} \operatorname {g2} \left (\left (\left (1-k \right ) \int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f} \left (k -1\right )}d \textit {\_f} +\left (k -1\right ) \int \operatorname {f2} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}\right )d \textit {\_f} +z^{1-m} {\mathrm e}^{\left (m -1\right ) \int \operatorname {g1} \left (x \right )d x}\right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.7.25.6 [1736] Problem 6
problem number 1736
Added June 27, 2019.
Problem Chapter 7.8.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x) y^k) w_y + (g_1(x) z+g_2(y) e^{\lambda z}) w_z = h_1(x)+h_2(y) \]
Mathematica ✗
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (f1[x]*y+f2[x]*y^k)*D[w[x, y,z], y] + (g1[x]*z+g2[y]*Exp[lambda*z])*D[w[x,y,z],z]== h1[x]+h2[y];
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)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+ (g1(x)*z+g2(y)*exp(lambda*z))*diff(w(x,y,z),z)= h1(x)+h2(y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.7.25.7 [1737] Problem 7
problem number 1737
Added June 27, 2019.
Problem Chapter 7.8.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) +f_2(x) e^{\lambda y}) w_y + (g_1(x) z+g_2(y) z^k) w_z = h_1(x)+h_2(y) \]
Mathematica ✗
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (f1[x]+f2[x]*Exp[lambda*y])*D[w[x, y,z], y] + (g1[x]*z+g2[y]*z^k)*D[w[x,y,z],z]== h1[x]+h2[y];
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)+ (f1(x)+f2(x)*exp(lambda*y))*diff(w(x,y,z),y)+ (g1(x)*z+g2(y)*z^k)*diff(w(x,y,z),z)= h1(x)+h2(y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\left (\operatorname {h1} \left (\textit {\_f} \right )+\operatorname {h2} \left (\frac {\ln \left (\frac {1}{-\int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} \lambda +\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda +{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}}\right )+\lambda \int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}{\lambda }\right )\right )d \textit {\_f} +f_{1} \left (\frac {-\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda -{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}}{\lambda }, \left (k -1\right ) \int _{}^{x}\operatorname {g2} \left (\frac {\ln \left (\frac {1}{-\int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} \lambda +\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda +{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}}\right )+\lambda \int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}{\lambda }\right ) {\mathrm e}^{\int \operatorname {g1} \left (\textit {\_f} \right )d \textit {\_f} \left (k -1\right )}d \textit {\_f} +z^{1-k} {\mathrm e}^{\int \operatorname {g1} \left (x \right )d x \left (k -1\right )}\right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.7.25.8 [1738] Problem 8
problem number 1738
Added June 27, 2019.
Problem Chapter 7.8.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) +f_2(x) e^{\lambda y}) w_y + (g_1(x) +g_2(y) e^{\beta z}) w_z = h_1(x)+h_2(y) \]
Mathematica ✗
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + (f1[x]+f2[x]*Exp[lambda*y])*D[w[x, y,z], y] + (g1[x]+g2[y]*Exp[beta*z])*D[w[x,y,z],z]== h1[x]+h2[y];
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)+ (f1(x)+f2(x)*exp(lambda*y))*diff(w(x,y,z),y)+ (g1(x)+g2(y)*exp(beta*z))*diff(w(x,y,z),z)= h1(x)+h2(y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\left (\operatorname {h1} \left (\textit {\_f} \right )+\operatorname {h2} \left (\frac {\ln \left (\frac {1}{-\int \operatorname {f2} \left (\textit {\_f} \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} \lambda +\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda +{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}}\right )+\lambda \int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}{\lambda }\right )\right )d \textit {\_f} +f_{1} \left (\frac {-{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}-\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda }{\lambda }, \frac {-\int _{}^{x}\operatorname {g2} \left (\frac {\ln \left (\frac {1}{-\int \operatorname {f2} \left (\textit {\_g} \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (\textit {\_g} \right )d \textit {\_g}}d \textit {\_g} \lambda +\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda +{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}}\right )+\lambda \int \operatorname {f1} \left (\textit {\_f} \right )d \textit {\_f}}{\lambda }\right ) {\mathrm e}^{\beta \int \operatorname {g1} \left (\textit {\_f} \right )d \textit {\_f}}d \textit {\_f} \beta -{\mathrm e}^{\beta \left (-z +\int \operatorname {g1} \left (x \right )d x \right )}}{\beta }\right )\]
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________