7.7.12 5.1

7.7.12.1 [1665] Problem 1

problem number 1665

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x)+\frac {c \log ^k(\lambda x) (-\log (\lambda x))^{-k} \operatorname {Gamma}(k+1,-\log (\lambda x))}{\lambda }+s x\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = s x +\int c \ln \left (\lambda x \right )^{k}d x +\textit {\_F1} \left (-a x +y , -b x +z \right )\] Contains unresolve integral because maple can not integrate \(\ln ^n(x)\)

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7.7.12.2 [1666] Problem 2

problem number 1666

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'));
 

\[w \left (x , y , z\right ) = \frac {a \textit {\_F1} \left (\frac {-a y +b x}{b}, \frac {-\left (\ln \left (\beta y \right )-1\right ) c \gamma y -b \Ei \left (1, -\ln \left (\gamma z \right )\right )}{c \gamma }\right )+\left (\ln \left (\alpha x \right )-1\right ) k x}{a}\]

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7.7.12.3 [1667] Problem 3

problem number 1667

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]];
 

\[\left \{\left \{w(x,y,z)\to x (c \log (\gamma x)-c+s)+c_1\left (y-\frac {a (-\log (\beta x))^{-n} \log ^n(\beta x) \operatorname {Gamma}(n+1,-\log (\beta x))}{\beta },z-\frac {b (-\log (\lambda x))^{-k} \log ^k(\lambda x) \operatorname {Gamma}(k+1,-\log (\lambda x))}{\lambda }\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = c x \ln \left (\gamma x \right )+\left (-c +s \right ) x +\textit {\_F1} \left (y -\left (\int a \ln \left (\beta x \right )^{n}d x \right ), z -\left (\int b \ln \left (\lambda x \right )^{k}d x \right )\right )\] Contains unresolve integral because maple can not integrate \(\ln ^n(x)\)

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7.7.12.4 [1668] Problem 4

problem number 1668

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]];
 

Failed

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'));
 

\[w \left (x , y , z\right ) = \int _{}^{x}\left (c \ln \left (\left (a \left (\int \ln \left (\textit {\_f} \lambda \right )^{n}d \textit {\_f} \right )+y -\left (\int a \ln \left (\lambda x \right )^{n}d x \right )\right ) \gamma \right )^{k}+s \ln \left (\left (b \left (\int \ln \left (\left (a \left (\int \ln \left (\textit {\_f} \lambda \right )^{n}d \textit {\_f} \right )+y -\left (\int a \ln \left (\lambda x \right )^{n}d x \right )\right ) \beta \right )^{m}d \textit {\_f} \right )+z -\left (\int _{}^{x}b \ln \left (\left (a \left (\int \ln \left (\textit {\_b} \lambda \right )^{n}d \textit {\_b} \right )+y -\left (\int a \ln \left (\lambda x \right )^{n}d x \right )\right ) \beta \right )^{m}d \textit {\_b} \right )\right ) \mu \right )^{r}\right )d \textit {\_f} +\textit {\_F1} \left (y -\left (\int a \ln \left (\lambda x \right )^{n}d x \right ), z -\left (\int _{}^{x}b \ln \left (\left (a \left (\int \ln \left (\textit {\_b} \lambda \right )^{n}d \textit {\_b} \right )+y -\left (\int a \ln \left (\lambda x \right )^{n}d x \right )\right ) \beta \right )^{m}d \textit {\_b} \right )\right )\]

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7.7.12.5 [1669] Problem 5

problem number 1669

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'));
 

\[w \left (x , y , z\right ) = \int _{}^{x}\frac {\left (\mathit {a2} \ln \left (\textit {\_f} \lambda 2 \right )^{\mathit {n2}}+\mathit {b2} \ln \left (\beta 2 \RootOf \left (\mathit {a1} \left (\int _{}^{\textit {\_Z}}\ln \left (\textit {\_a} \beta 1 \right )^{-\mathit {m1}}d \textit {\_a} \right )-\mathit {b1} \left (\int \ln \left (\textit {\_f} \lambda 1 \right )^{-\mathit {n1}}d \textit {\_f} \right )+\mathit {b1} \left (\int \ln \left (\lambda 1 x \right )^{-\mathit {n1}}d x \right )-\mathit {b1} \left (\int \frac {\mathit {a1} \ln \left (\beta 1 y \right )^{-\mathit {m1}}}{\mathit {b1}}d y \right )\right )\right )^{\mathit {m2}}+\mathit {c2} \ln \left (\gamma 2 \RootOf \left (\mathit {a1} \left (\int _{}^{\textit {\_Z}}\ln \left (\textit {\_a} \gamma 1 \right )^{-\mathit {k1}}d \textit {\_a} \right )-\mathit {c1} \left (\int \ln \left (\textit {\_f} \lambda 1 \right )^{-\mathit {n1}}d \textit {\_f} \right )+\mathit {c1} \left (\int \ln \left (\lambda 1 x \right )^{-\mathit {n1}}d x \right )-\mathit {c1} \left (\int \frac {\mathit {a1} \ln \left (\gamma 1 z \right )^{-\mathit {k1}}}{\mathit {c1}}d z \right )\right )\right )^{\mathit {k2}}\right ) \ln \left (\textit {\_f} \lambda 1 \right )^{-\mathit {n1}}}{\mathit {a1}}d \textit {\_f} +\textit {\_F1} \left (-\left (\int \ln \left (\lambda 1 x \right )^{-\mathit {n1}}d x \right )+\int \frac {\mathit {a1} \ln \left (\beta 1 y \right )^{-\mathit {m1}}}{\mathit {b1}}d y , -\left (\int \ln \left (\lambda 1 x \right )^{-\mathit {n1}}d x \right )+\int \frac {\mathit {a1} \ln \left (\gamma 1 z \right )^{-\mathit {k1}}}{\mathit {c1}}d z \right )\] Contains RootOf and unresolved integrals \(\ln ^n(x)\)

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