Added June 27, 2019.
Problem Chapter 7.8.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + f(x) w_y + g(x) w_z = h_2(x) y+h_1(x) + h_0(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + f[x]*D[w[x, y,z], y] + g[x]*D[w[x,y,z],z]== h2[x]*y+h1[x]*z+h0[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (\text {h0}(K[3])+\text {h2}(K[3]) \left (y-\int _1^xf(K[1])dK[1]+\int _1^{K[3]}f(K[1])dK[1]\right )+\text {h1}(K[3]) \left (z-\int _1^xg(K[2])dK[2]+\int _1^{K[3]}g(K[2])dK[2]\right )\right )dK[3]+c_1\left (y-\int _1^xf(K[1])dK[1],z-\int _1^xg(K[2])dK[2]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ f(x)*diff(w(x,y,z),y)+ g(x)*diff(w(x,y,z),z)= h2(x)*y+h1(x)*z+h0(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\left (\left (\int f \left (\mathit {\_f} \right )d \mathit {\_f} \right ) \mathit {h2} \left (\mathit {\_f} \right )+\left (\int g \left (\mathit {\_f} \right )d \mathit {\_f} \right ) \mathit {h1} \left (\mathit {\_f} \right )+\mathit {h0} \left (\mathit {\_f} \right )+\left (z -\left (\int g \left (x \right )d x \right )\right ) \mathit {h1} \left (\mathit {\_f} \right )+\left (y -\left (\int f \left (x \right )d x \right )\right ) \mathit {h2} \left (\mathit {\_f} \right )\right )d\mathit {\_f} +\mathit {\_F1} \left (y -\left (\int f \left (x \right )d x \right ), z -\left (\int g \left (x \right )d x \right )\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + f(x)(y+a) w_y + g(x) (z+b)w_z = h(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + f[x]*(y+a)*D[w[x, y,z], y] + g[x]*(z+b)*D[w[x,y,z],z]== h[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^xh(K[5])dK[5]+c_1\left (y \exp \left (-\int _1^xf(K[1])dK[1]\right )-\int _1^xa \exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right ) f(K[2])dK[2],z \exp \left (-\int _1^xg(K[3])dK[3]\right )-\int _1^xb \exp \left (-\int _1^{K[4]}g(K[3])dK[3]\right ) g(K[4])dK[4]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ f(x)*(y+a)*diff(w(x,y,z),y)+ g(x)*(z+b)*diff(w(x,y,z),z)= h(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int h \left (x \right )d x +\mathit {\_F1} \left (\left (a +y \right ) {\mathrm e}^{-\left (\int f \left (x \right )d x \right )}, \left (b +z \right ) {\mathrm e}^{-\left (\int g \left (x \right )d x \right )}\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a y+f(x)) w_y + (b z+g(x))w_z = h(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a*y+f[x])*D[w[x, y,z], y] + (b*z+g[x])*D[w[x,y,z],z]== h[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^xh(K[3])dK[3]+c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} f(K[1])dK[1],z e^{-b x}-\int _1^xe^{-b K[2]} g(K[2])dK[2]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ (a*y+f(x))*diff(w(x,y,z),y)+ (b*z+g(x))*diff(w(x,y,z),z)= h(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int h \left (x \right )d x +\mathit {\_F1} \left (y \,{\mathrm e}^{-a x}-\left (\int {\mathrm e}^{-a x} f \left (x \right )d x \right ), z \,{\mathrm e}^{-b x}-\left (\int {\mathrm e}^{-b x} g \left (x \right )d x \right )\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x)) w_y + (g_1(x) y+g_2(x))w_z = h_2(x) y + h_1(x) z +h_0(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]*y+f2[x])*D[w[x, y,z], y] + (g1[x]*y+g2[x])*D[w[x,y,z],z]== h2[x]*y+h1[x]*z+h0[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) \left (\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \text {h0}(K[5])+\exp \left (\int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) \text {h2}(K[5]) \left (y-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \int _1^x\exp \left (-\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3]+\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \int _1^{K[5]}\exp \left (-\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3]\right )+\text {h1}(K[5]) \left (-y \int _1^x\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]+\left (y-\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \int _1^x\exp \left (-\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3]+\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \int _1^{K[5]}\exp \left (-\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3]\right ) \int _1^{K[5]}\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]+\exp \left (\int _1^x\text {f1}(K[1])dK[1]\right ) \left (z-\int _1^x\left (\text {g2}(K[4])-\exp \left (-\int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[4]) \int _1^{K[4]}\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]\right )dK[4]+\int _1^{K[5]}\left (\text {g2}(K[4])-\exp \left (-\int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[4]) \int _1^{K[4]}\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]\right )dK[4]\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[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3],-\int _1^x\left (\text {g2}(K[4])-\exp \left (-\int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[4]) \int _1^{K[4]}\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]\right )dK[4]-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 {g1}(K[2])dK[2]+z\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)*y+g2(x))*diff(w(x,y,z),z)= h2(x)*y+h1(x)*z+h0(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 _{}^{x}\left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}} \mathit {h2} \left (\mathit {\_g} \right )+\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {f2} \left (\mathit {\_g} \right )d \mathit {\_g} \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}} \mathit {h2} \left (\mathit {\_g} \right )-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}} \mathit {h2} \left (\mathit {\_g} \right )+z \mathit {h1} \left (\mathit {\_g} \right )+\left (\int \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}} \mathit {g1} \left (\mathit {\_g} \right )+\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g} \right )} \mathit {f2} \left (\mathit {\_g} \right )d \mathit {\_g} \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}} \mathit {g1} \left (\mathit {\_g} \right )-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_g} \right )d \mathit {\_g}} \mathit {g1} \left (\mathit {\_g} \right )+\mathit {g2} \left (\mathit {\_g} \right )\right )d \mathit {\_g} \right ) \mathit {h1} \left (\mathit {\_g} \right )-\left (\int _{}^{x}\left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )+\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )+\mathit {g2} \left (\mathit {\_f} \right )\right )d\mathit {\_f} \right ) \mathit {h1} \left (\mathit {\_g} \right )+\mathit {h0} \left (\mathit {\_g} \right )\right )d\mathit {\_g} +\mathit {\_F1} \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), z -\left (\int _{}^{x}\left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )+\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )+\mathit {g2} \left (\mathit {\_f} \right )\right )d\mathit {\_f} \right )\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x)) w_y + (g_1(x) z+g_2(x))w_z = h_2(x) y + h_1(x) z +h_0(x) \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]*z+f2[x])*D[w[x, y,z], y] + (g1[x]*y+g2[x])*D[w[x,y,z],z]== h2[x]*y+h1[x]*z+h0[x]; 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)*z+f2(x))*diff(w(x,y,z),y)+ (g1(x)*y+g2(x))*diff(w(x,y,z),z)= h2(x)*y+h1(x)*z+h0(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added June 27, 2019.
Problem Chapter 7.8.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (y^2-a^2+a \lambda \sinh (\lambda x)-a^2 \sinh ^2(\lambda x)) w_y + f(x) \sinh (\gamma z) w_z = g(x) \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (y^2-a^2 + a *lambda*Sin[lambda*x]-a^2*Sinh[lambda*x]^2)*D[w[x, y,z], y] + f[x]*Sinh[gamma*z]*D[w[x,y,z],z]== g[x]; 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^2 + a *lambda*sin(lambda*x)-a^2*sinh(lambda*x)^2)*diff(w(x,y,z),y)+ f(x)*sinh(gamma*z)*diff(w(x,y,z),z)= g(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
\[w \left (x , y , z\right )=c_{1}-\frac {2 \mathit {\_c}_{3} \arctanh \left ({\mathrm e}^{\gamma z}\right )}{\gamma }+\mathit {\_F1} \left (x \right )+\mathit {\_F2} \left (y \right )\boldsymbol {\mathrm {where}}\left [\left \{-\frac {a^{2} \cosh \left (2 \lambda x \right ) \left (\frac {d}{d y}\mathit {\_F2} \left (y \right )\right )}{2}+a \lambda \left (\frac {d}{d y}\mathit {\_F2} \left (y \right )\right ) \sin \left (\lambda x \right )-\frac {a^{2} \left (\frac {d}{d y}\mathit {\_F2} \left (y \right )\right )}{2}+y^{2} \left (\frac {d}{d y}\mathit {\_F2} \left (y \right )\right )+\mathit {\_c}_{3} f \left (x \right )+\frac {d}{d x}\mathit {\_F1} \left (x \right )-g \left (x \right )=0\right \}\right ]\] Gives Warning: Incomplete separation
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Added June 27, 2019.
Problem Chapter 7.8.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y + f_2(x) y^k) w_y + (g_1(x) z+g_2(x) z^m) w_z = h(x) \]
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[x]*z^m)*D[w[x,y,z],z]== h[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^xh(K[5])dK[5]+c_1\left ((k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\text {f1}(K[1])dK[1]\right ),(m-1) \int _1^x\exp \left ((m-1) \int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]+z^{1-m} \exp \left ((m-1) \int _1^x\text {g1}(K[3])dK[3]\right )\right )\right \}\right \}\]
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(x)*z^m)*diff(w(x,y,z),z)= h(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int h \left (x \right )d x +\mathit {\_F1} \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), z^{-m +1} {\mathrm e}^{\left (m -1\right ) \left (\int \mathit {g1} \left (x \right )d x \right )}+\left (m -1\right ) \left (\int {\mathrm e}^{\left (m -1\right ) \left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {g2} \left (x \right )d x \right )\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y + f_2(x) y^k) w_y + (g_1(x) +g_2(x) e^{\lambda z}) w_z = h(x) \]
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]+g2[x]*Exp[lambda*z])*D[w[x,y,z],z]== h[x]; 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)+g2(x)*exp(lambda*z))*diff(w(x,y,z),z)= h(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int h \left (x \right )d x +\mathit {\_F1} \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), \frac {-\lambda \left (\int {\mathrm e}^{\lambda \left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {g2} \left (x \right )d x \right )-{\mathrm e}^{\left (-z +\int \mathit {g1} \left (x \right )d x \right ) \lambda }}{\lambda }\right )\]
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Added June 27, 2019.
Problem Chapter 7.8.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x)+ f_2(x) e^{\lambda y}) w_y + (g_1(x) +g_2(x) e^{\beta z}) w_z = h(x) \]
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[x]*Exp[beta*z])*D[w[x,y,z],z]== h[x]; 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(x)*exp(beta*z))*diff(w(x,y,z),z)= h(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int h \left (x \right )d x +\mathit {\_F1} \left (\frac {-\lambda \left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )-{\mathrm e}^{\left (-y +\int \mathit {f1} \left (x \right )d x \right ) \lambda }}{\lambda }, \frac {-\beta \left (\int {\mathrm e}^{\beta \left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {g2} \left (x \right )d x \right )-{\mathrm e}^{\left (-z +\int \mathit {g1} \left (x \right )d x \right ) \beta }}{\beta }\right )\]
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