6.8.23 8.1
6.8.23.1 [1891] Problem 1
problem number 1891
Added December 1, 2019.
Problem Chapter 8.8.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + f(x) w_y + g(x) w_z = \left ( h_2(x) y+h_1(x) z+h_0(x) \right ) w \]
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])*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\int _1^xf(K[1])dK[1],z-\int _1^xg(K[2])dK[2]\right ) \exp \left (\int _1^x\left (\text {h0}(K[3])+\text {h2}(K[3]) \left (y-\int _1^xf(K[1])dK[1]+\int _1^{K[3]}f(K[1])dK[1]\right )+\text {h1}(K[3]) \left (z-\int _1^xg(K[2])dK[2]+\int _1^{K[3]}g(K[2])dK[2]\right )\right )dK[3]\right )\right \}\right \}\]
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))*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-\int f \left (x \right )d x +y , -\int g \left (x \right )d x +z \right ) {\mathrm e}^{\int _{}^{x}\left (-\operatorname {h1} \left (\textit {\_f} \right ) \int g \left (x \right )d x +\operatorname {h2} \left (\textit {\_f} \right ) \int f \left (\textit {\_f} \right )d \textit {\_f} -\operatorname {h2} \left (\textit {\_f} \right ) \int f \left (x \right )d x +\operatorname {h2} \left (\textit {\_f} \right ) y +\operatorname {h1} \left (\textit {\_f} \right ) \int g \left (\textit {\_f} \right )d \textit {\_f} +\operatorname {h1} \left (\textit {\_f} \right ) z +\operatorname {h0} \left (\textit {\_f} \right )\right )d \textit {\_f}}\]
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6.8.23.2 [1892] Problem 2
problem number 1892
Added December 1, 2019.
Problem Chapter 8.8.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + f(x)(y+a) w_y + g(x)(z+b) w_z = h(x) w \]
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]*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^xh(K[5])dK[5]\right ) c_1\left (y \exp \left (-\int _1^xf(K[1])dK[1]\right )-\int _1^xa \exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right ) f(K[2])dK[2],z \exp \left (-\int _1^xg(K[3])dK[3]\right )-\int _1^xb \exp \left (-\int _1^{K[4]}g(K[3])dK[3]\right ) g(K[4])dK[4]\right )\right \}\right \}\]
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)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\left (y +a \right ) {\mathrm e}^{-\int f \left (x \right )d x}, \left (z +b \right ) {\mathrm e}^{-\int g \left (x \right )d x}\right ) {\mathrm e}^{\int h \left (x \right )d x}\]
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6.8.23.3 [1893] Problem 3
problem number 1893
Added December 1, 2019.
Problem Chapter 8.8.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( a y+f(x) \right ) w_y + \left ( b z+g(x) \right ) w_z = h(x) w \]
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]*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^xh(K[3])dK[3]\right ) c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} f(K[1])dK[1],z e^{-b x}-\int _1^xe^{-b K[2]} g(K[2])dK[2]\right )\right \}\right \}\]
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)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-\int f \left (x \right ) {\mathrm e}^{-a x}d x +{\mathrm e}^{-a x} y , {\mathrm e}^{-b x} z -\int g \left (x \right ) {\mathrm e}^{-b x}d x \right ) {\mathrm e}^{\int h \left (x \right )d x}\]
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6.8.23.4 [1894] Problem 4
problem number 1894
Added December 1, 2019.
Problem Chapter 8.8.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( f_1(x) y+ f_2(x) \right ) w_y + \left ( g_1(x) y+ g_2(x) \right ) w_z = \left ( h_2(x) y+h_1(x) z+h_0(x) \right ) w \]
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])*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 \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 ) \exp \left (\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]\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))*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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6.8.23.5 [1895] Problem 5
problem number 1895
Added December 1, 2019.
Problem Chapter 8.8.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( f_1(x) y+ f_2(x) \right ) w_y + \left ( g_1(x) z+ g_2(x) \right ) w_z = \left ( h_2(x) y+h_1(x) z+h_0(x) \right ) w \]
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[x])*D[w[x,y,z],z]==(h2[x]*y+h1[x]*z+h0[x])*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2],z \exp \left (-\int _1^x\text {g1}(K[3])dK[3]\right )-\int _1^x\exp \left (-\int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]\right ) \exp \left (\int _1^x\left (\text {h0}(K[5])+\exp \left (\int _1^{K[5]}\text {f1}(K[1])dK[1]\right ) \text {h2}(K[5]) \left (\exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) y-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+\int _1^{K[5]}\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]\right )+\exp \left (\int _1^{K[5]}\text {g1}(K[3])dK[3]\right ) \text {h1}(K[5]) \left (\exp \left (-\int _1^x\text {g1}(K[3])dK[3]\right ) z-\int _1^x\exp \left (-\int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]+\int _1^{K[5]}\exp \left (-\int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]\right )\right )dK[5]\right )\right \}\right \}\]
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(x))*diff(w(x,y,z),z)= (h2(x)*y+h1(x)*z+h0(x))*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left ({\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x} y -\int \operatorname {f2} \left (x \right ) {\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x}d x , -\int \operatorname {g2} \left (x \right ) {\mathrm e}^{-\int \operatorname {g1} \left (x \right )d x}d x +{\mathrm e}^{-\int \operatorname {g1} \left (x \right )d x} z \right ) {\mathrm e}^{\int _{}^{x}\left (\operatorname {h1} \left (\textit {\_h} \right ) z \,{\mathrm e}^{\int \operatorname {g1} \left (\textit {\_h} \right )d \textit {\_h} -\int \operatorname {g1} \left (x \right )d x}+\operatorname {h2} \left (\textit {\_h} \right ) y \,{\mathrm e}^{\int \operatorname {f1} \left (\textit {\_h} \right )d \textit {\_h} -\int \operatorname {f1} \left (x \right )d x}+\operatorname {h1} \left (\textit {\_h} \right ) {\mathrm e}^{\int \operatorname {g1} \left (\textit {\_h} \right )d \textit {\_h}} \int \operatorname {g2} \left (\textit {\_h} \right ) {\mathrm e}^{-\int \operatorname {g1} \left (\textit {\_h} \right )d \textit {\_h}}d \textit {\_h} -{\mathrm e}^{\int \operatorname {g1} \left (\textit {\_h} \right )d \textit {\_h}} \operatorname {h1} \left (\textit {\_h} \right ) \int \operatorname {g2} \left (x \right ) {\mathrm e}^{-\int \operatorname {g1} \left (x \right )d x}d x +\operatorname {h2} \left (\textit {\_h} \right ) {\mathrm e}^{\int \operatorname {f1} \left (\textit {\_h} \right )d \textit {\_h}} \int \operatorname {f2} \left (\textit {\_h} \right ) {\mathrm e}^{-\int \operatorname {f1} \left (\textit {\_h} \right )d \textit {\_h}}d \textit {\_h} -{\mathrm e}^{\int \operatorname {f1} \left (\textit {\_h} \right )d \textit {\_h}} \operatorname {h2} \left (\textit {\_h} \right ) \int \operatorname {f2} \left (x \right ) {\mathrm e}^{-\int \operatorname {f1} \left (x \right )d x}d x +\operatorname {h0} \left (\textit {\_h} \right )\right )d \textit {\_h}}\]
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6.8.23.6 [1896] Problem 6
problem number 1896
Added December 1, 2019.
Problem Chapter 8.8.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( y^2-a^2+a \lambda \sinh (\lambda x)-a^2 \sinh ^2(\lambda x) \right ) w_y + f(x) \sinh (\gamma z) w_z = g(x) w \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x,y,z],x]+(y^2-a^2+a*lambda*Sinh[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]*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\begin{align*}& \left \{w(x,y,z)\to \exp \left (\int _1^xg(K[3])dK[3]\right ) c_1\left (-\frac {\gamma \int _1^xf(K[2])dK[2]+\text {arctanh}(\cosh (\gamma z))}{\gamma },\frac {2 \lambda e^{\frac {a e^{-\lambda x} \left (e^{2 \lambda x}-1\right )}{\lambda }+\lambda x}}{a e^{2 \lambda x}+a-2 y e^{\lambda x}}-\int _1^{e^{\lambda x}}\frac {e^{\frac {a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]}dK[1]\right )\right \}\\& \left \{w(x,y,z)\to \exp \left (\int _1^xg(K[4])dK[4]\right ) c_1\left (-\frac {\gamma \int _1^xf(K[2])dK[2]+\text {arctanh}(\cosh (\gamma z))}{\gamma },\frac {2 \lambda e^{\frac {a e^{-\lambda x} \left (e^{2 \lambda x}-1\right )}{\lambda }+\lambda x}}{a e^{2 \lambda x}+a-2 y e^{\lambda x}}-\int _1^{e^{\lambda x}}\frac {e^{\frac {a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]}dK[1]\right )\right \}\\\end{align*}
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ (y^2-a^2+a*lambda*sinh(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)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {\left (\left (-2 \cosh \left (\lambda x \right ) a -2 y \right ) \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, \frac {-8 i a +3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )+i \operatorname {HeunCPrime}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, \frac {-8 i a +3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cosh \left (\lambda x \right ) \lambda \right ) \sqrt {\sinh \left (\lambda x \right )+i}}{\left (\left (2 i a +2 \sinh \left (\lambda x \right ) a +\lambda \right ) \cosh \left (\lambda x \right )+2 y \left (\sinh \left (\lambda x \right )+i\right )\right ) \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, \frac {-8 i a +3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )-\left (i \sinh \left (\lambda x \right )-1\right ) \lambda \cosh \left (\lambda x \right ) \operatorname {HeunCPrime}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, \frac {-8 i a +3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )}, \frac {-\int f \left (x \right )d x \gamma -2 \,\operatorname {arctanh}\left ({\mathrm e}^{\gamma z}\right )}{\gamma }\right ) {\mathrm e}^{\int g \left (x \right )d x}\]
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6.8.23.7 [1897] Problem 7
problem number 1897
Added December 1, 2019.
Problem Chapter 8.8.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( f_1(x) y+f_2(x) y^k \right ) w_y + \left ( g_1(x) z+g_2(x) z^m \right ) w_z = h(x) w \]
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]*w[x,y,z];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^xh(K[5])dK[5]\right ) c_1\left ((k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\text {f1}(K[1])dK[1]\right ),(m-1) \int _1^x\exp \left ((m-1) \int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]+z^{1-m} \exp \left ((m-1) \int _1^x\text {g1}(K[3])dK[3]\right )\right )\right \}\right \}\]
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)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\left (k -1\right ) \int \operatorname {f2} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}, \left (m -1\right ) \int \operatorname {g2} \left (x \right ) {\mathrm e}^{\left (m -1\right ) \int \operatorname {g1} \left (x \right )d x}d x +z^{1-m} {\mathrm e}^{\left (m -1\right ) \int \operatorname {g1} \left (x \right )d x}\right ) {\mathrm e}^{\int h \left (x \right )d x}\]
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6.8.23.8 [1898] Problem 8
problem number 1898
Added December 1, 2019.
Problem Chapter 8.8.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( f_1(x) y+f_2(x) y^k \right ) w_y + \left ( g_1(x) +g_2(x) e^{\lambda z} \right ) w_z = h(x) w \]
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]*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)+ ( 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)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\left (k -1\right ) \int \operatorname {f2} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \operatorname {f1} \left (x \right )d x}, \frac {-\int \operatorname {g2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {g1} \left (x \right )d x}d x \lambda -{\mathrm e}^{\lambda \left (-z +\int \operatorname {g1} \left (x \right )d x \right )}}{\lambda }\right ) {\mathrm e}^{\int h \left (x \right )d x}\]
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6.8.23.9 [1899] Problem 9
problem number 1899
Added December 1, 2019.
Problem Chapter 8.8.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + \left ( f_1(x)+f_2(x) e^{\lambda y} \right ) w_y + \left ( g_1(x) +g_2(x) e^{\beta z} \right ) w_z = h(x) w \]
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]*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)+ ( 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)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {-\int \operatorname {f2} \left (x \right ) {\mathrm e}^{\lambda \int \operatorname {f1} \left (x \right )d x}d x \lambda -{\mathrm e}^{\lambda \left (-y +\int \operatorname {f1} \left (x \right )d x \right )}}{\lambda }, \frac {-\int \operatorname {g2} \left (x \right ) {\mathrm e}^{\beta \int \operatorname {g1} \left (x \right )d x}d x \beta -{\mathrm e}^{\beta \left (-z +\int \operatorname {g1} \left (x \right )d x \right )}}{\beta }\right ) {\mathrm e}^{\int h \left (x \right )d x}\]
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