6.7.12 5.1
6.7.12.1 [1663] Problem 1
problem number 1663
Added June 26, 2019.
Problem Chapter 7.5.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \ln ^k(\lambda x)+s \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*D[w[x, y,z], y] + b*D[w[x,y,z],z]== c*Log[lambda*x]^k+s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \log ^k(\lambda K[1])+s\right )dK[1]+c_1(y-a x,z-b x)\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*diff(w(x,y,z),y)+ b*diff(w(x,y,z),z)=c*ln(lambda*x)^k+s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = c \int \ln \left (\lambda x \right )^{k}d x +s x +f_{1} \left (-a x +y , -b x +z \right )\]
Contains unresolve integral because maple can not integrate
\(\ln ^n(x)\) ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.7.12.2 [1664] Problem 2
problem number 1664
Added June 26, 2019.
Problem Chapter 7.5.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \ln (\beta y) \ln (\gamma z) w_z = k \ln (\alpha x) \]
Mathematica ✗
ClearAll["Global`*"];
pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Log[beta*y]*Log[gamma*z]*D[w[x,y,z],z]== k*Log[alpha*x];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*ln(beta*y)*ln(gamma*z)*diff(w(x,y,z),z)=k*ln(alpha*x);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {f_{1} \left (\frac {-y a +b x}{b}, \frac {-b \,\operatorname {Ei}_{1}\left (-\ln \left (\gamma z \right )\right )-y c \gamma \left (\ln \left (\beta y \right )-1\right )}{c \gamma }\right ) a +k x \left (\ln \left (\alpha x \right )-1\right )}{a}\]
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6.7.12.3 [1665] Problem 3
problem number 1665
Added June 26, 2019.
Problem Chapter 7.5.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \ln ^n(\beta x) w_y + b \ln ^k(\lambda x) w_z = c \ln ^m(\gamma x)+s \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Log[beta*x]^n*D[w[x, y,z], y] + b*Log[lambda*x]^k*D[w[x,y,z],z]== c*Log[gamma*x]+s;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to x (c \log (\gamma x)-c+s)+c_1\left (y-\int _1^xa \log ^n(\beta K[1])dK[1],z-\int _1^xb \log ^k(\lambda K[2])dK[2]\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*ln(beta*x)^n*diff(w(x,y,z),y)+ b*ln(lambda*x)^k*diff(w(x,y,z),z)=c*ln(gamma*x)+s;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-a \int \ln \left (\beta x \right )^{n}d x +y , -b \int \ln \left (\lambda x \right )^{k}d x +z \right )+c x \ln \left (\gamma x \right )+\left (-c +s \right ) x\]
Contains unresolve integral because maple can not integrate
\(\ln ^n(x)\) ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
6.7.12.4 [1666] Problem 4
problem number 1666
Added June 26, 2019.
Problem Chapter 7.5.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \ln ^n(\lambda x) w_y + b \ln ^m(\beta y) w_z = c \ln ^k(\gamma y)+s \ln ^r(\mu z) \]
Mathematica ✓
ClearAll["Global`*"];
pde = D[w[x, y,z], x] + a*Log[lambda*x]^n*D[w[x, y,z], y] + b*Log[beta*y]^m*D[w[x,y,z],z]== c*Log[gamma*y]^k+s*Log[mu*z]^r;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \log ^k\left (\gamma \left (y-\int _1^xa \log ^n(\lambda K[1])dK[1]+\int _1^{K[3]}a \log ^n(\lambda K[1])dK[1]\right )\right )+s \log ^r\left (\mu \left (z-\int _1^xb \log ^m\left (\beta \left (y-\int _1^xa \log ^n(\lambda K[1])dK[1]+\int _1^{K[2]}a \log ^n(\lambda K[1])dK[1]\right )\right )dK[2]+\int _1^{K[3]}b \log ^m\left (\beta \left (y-\int _1^xa \log ^n(\lambda K[1])dK[1]+\int _1^{K[2]}a \log ^n(\lambda K[1])dK[1]\right )\right )dK[2]\right )\right )\right )dK[3]+c_1\left (y-\int _1^xa \log ^n(\lambda K[1])dK[1],z-\int _1^xb \log ^m\left (\beta \left (y-\int _1^xa \log ^n(\lambda K[1])dK[1]+\int _1^{K[2]}a \log ^n(\lambda K[1])dK[1]\right )\right )dK[2]\right )\right \}\right \}\]
Maple ✓
restart;
local gamma;
pde := diff(w(x,y,z),x)+ a*ln(lambda*x)^n*diff(w(x,y,z),y)+ b*ln(beta*y)^m*diff(w(x,y,z),z)=c*ln(gamma*y)^k+s*ln(mu*z)^r;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\left (c {\ln \left (\gamma \left (a \int \ln \left (\lambda \textit {\_f} \right )^{n}d \textit {\_f} -a \int \ln \left (\lambda x \right )^{n}d x +y \right )\right )}^{k}+s {\ln \left (\mu \left (b \int {\ln \left (\beta \left (a \int \ln \left (\lambda \textit {\_f} \right )^{n}d \textit {\_f} -a \int \ln \left (\lambda x \right )^{n}d x +y \right )\right )}^{m}d \textit {\_f} -b \int _{}^{x}{\ln \left (\beta \left (a \int \ln \left (\lambda \textit {\_f} \right )^{n}d \textit {\_f} -a \int \ln \left (\lambda x \right )^{n}d x +y \right )\right )}^{m}d \textit {\_f} +z \right )\right )}^{r}\right )d \textit {\_f} +f_{1} \left (-a \int \ln \left (\lambda x \right )^{n}d x +y , -b \int _{}^{x}{\ln \left (\beta \left (a \int \ln \left (\lambda \textit {\_f} \right )^{n}d \textit {\_f} -a \int \ln \left (\lambda x \right )^{n}d x +y \right )\right )}^{m}d \textit {\_f} +z \right )\]
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6.7.12.5 [1667] Problem 5
problem number 1667
Added June 26, 2019.
Problem Chapter 7.5.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 \ln ^{n_1}(\lambda _1 x) w_x + b_1 \ln ^{m_1}(\beta _1 y) w_y + c_1 \ln ^{k_1}(\gamma _1 z) w_z = a_2 \ln ^{n_2}(\lambda _2 x)+ b_2 \ln ^{m_2}(\beta _2 y) + c_2 \ln ^{k_2}(\gamma _2 z) \]
Mathematica ✗
ClearAll["Global`*"];
pde = a1*Log[lambda1*x]^n1*D[w[x, y,z], x] + b1*Log[beta1*y]^m1*D[w[x, y,z], y] + c1*Log[gamma1*z]^k1*D[w[x,y,z],z]== a2*Log[lambda2*x]^n2*D[w[x, y,z], x] + b2*Log[beta2*y]^m2+ c2*Log[gamma2*z]^k2;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart;
local gamma;
pde := a1*ln(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*ln(beta1*y)^m1*diff(w(x,y,z),y)+ c1*ln(gamma1*z)^k1*diff(w(x,y,z),z)=a2*ln(lambda2*x)^n2+ b2*ln(beta2*y)^m2+ c2*ln(gamma2*z)^k2;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\int _{}^{x}\left (\operatorname {a2} \ln \left (\lambda \operatorname {2} \textit {\_f} \right )^{\operatorname {n2}}+\operatorname {b2} {\ln \left (\beta \operatorname {2} \operatorname {RootOf}\left (\operatorname {a1} \int \ln \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y +\int \ln \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f} \operatorname {b1} -\operatorname {a1} \int _{}^{\textit {\_Z}}\ln \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} -\int \ln \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x \operatorname {b1} \right )\right )}^{\operatorname {m2}}+\operatorname {c2} {\ln \left (\gamma \operatorname {2} \operatorname {RootOf}\left (\int \ln \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f} \operatorname {c1} +\operatorname {a1} \int \ln \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z -\operatorname {a1} \int _{}^{\textit {\_Z}}\ln \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} -\int \ln \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x \operatorname {c1} \right )\right )}^{\operatorname {k2}}\right ) \ln \left (\lambda \operatorname {1} \textit {\_f} \right )^{-\operatorname {n1}}d \textit {\_f}}{\operatorname {a1}}+f_{1} \left (-\int \ln \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x +\frac {\operatorname {a1} \int \ln \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y}{\operatorname {b1}}, -\int \ln \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x +\frac {\operatorname {a1} \int \ln \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z}{\operatorname {c1}}\right )\]
Contains RootOf and unresolved integrals
\(\ln ^n(x)\) ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________