6.5.12 5.1

6.5.12.1 [1276] Problem 1

problem number 1276

Added April 5, 2019.

Problem Chapter 5.5.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = c w + \ln ^k(\lambda x) \ln ^n(\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ Log[lambda*x]^k*Log[beta*y]^n; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \log ^k(\lambda K[1]) \log ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ln(lambda*x)^k*ln(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}\ln \left (\lambda \textit {\_a} \right )^{k} \ln \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \beta }{a}\right )^{n} {\mathrm e}^{-\frac {c \textit {\_a}}{a}}d \textit {\_a}}{a}+f_{1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]

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6.5.12.2 [1277] Problem 2

problem number 1277

Added April 5, 2019.

Problem Chapter 5.5.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = c \ln ^k(\lambda x) w+ s \ln ^n(\beta x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*Log[lambda*x]^k*w[x,y]+s*Log[beta*x]^n; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \log ^k(\lambda K[1])}{a}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {c \log ^k(\lambda K[1])}{a}dK[1]\right ) s \log ^n(\beta K[2])}{a}dK[2]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*ln(lambda*x)^k*w(x,y)+s*ln(beta*x)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\frac {s \int \ln \left (\beta x \right )^{n} {\mathrm e}^{-\frac {c \int \ln \left (\lambda x \right )^{k}d x}{a}}d x}{a}+f_{1} \left (y -\frac {b x}{a}\right )\right ) {\mathrm e}^{\frac {c \int \ln \left (\lambda x \right )^{k}d x}{a}}\]

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6.5.12.3 [1278] Problem 3

problem number 1278

Added April 5, 2019.

Problem Chapter 5.5.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = \left ( c_1 \ln ^{n_1}(\lambda _1 x) +c_2 \ln ^{n_2}(\lambda _2 y) \right ) w + s_1 \ln ^{k_1}(\beta _1 x)+s_2 \ln ^{k_2}(\beta _2 y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == ( c1*Log[lambda1*x]^n1 +c2*Log[lambda2*y]^n2)*w[x,y] + s1*Log[beta1*x]^k1+s2*Log[beta2*y]*k2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {\text {c1} \log ^{\text {n1}}(\text {lambda1} K[1])+\text {c2} \log ^{\text {n2}}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {c1} \log ^{\text {n1}}(\text {lambda1} K[1])+\text {c2} \log ^{\text {n2}}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]\right ) \left (\text {s1} \log ^{\text {k1}}(\text {beta1} K[2])+\text {k2} \text {s2} \log \left (\text {beta2} \left (y+\frac {b (K[2]-x)}{a}\right )\right )\right )}{a}dK[2]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = ( c1*ln(lambda1*x)^n1 +c2*ln(lambda2*y)^n2)*w(x,y) + s1*ln(beta1*x)^k1+s2*ln(beta2*y)*k2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}{\mathrm e}^{-\frac {\int \left (\operatorname {c1} \ln \left (\lambda \operatorname {1} \textit {\_a} \right )^{\operatorname {n1}}+\operatorname {c2} \ln \left (\frac {\left (a y -b \left (-\textit {\_a} +x \right )\right ) \lambda \operatorname {2} }{a}\right )^{\operatorname {n2}}\right )d \textit {\_a}}{a}} \left (\operatorname {s2} \ln \left (\frac {\left (a y -b \left (-\textit {\_a} +x \right )\right ) \beta \operatorname {2} }{a}\right ) \operatorname {k2} +\operatorname {s1} \ln \left (\beta \operatorname {1} \textit {\_a} \right )^{\operatorname {k1}}\right )d \textit {\_a}}{a}+f_{1} \left (y -\frac {b x}{a}\right )\right ) {\mathrm e}^{\frac {\int _{}^{x}\left (\operatorname {c1} \ln \left (\lambda \operatorname {1} \textit {\_a} \right )^{\operatorname {n1}}+\operatorname {c2} \ln \left (\frac {\left (a y -b \left (-\textit {\_a} +x \right )\right ) \lambda \operatorname {2} }{a}\right )^{\operatorname {n2}}\right )d \textit {\_a}}{a}}\]

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6.5.12.4 [1279] Problem 4

problem number 1279

Added April 5, 2019.

Problem Chapter 5.5.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a \ln (\lambda x) w_x + b \ln (\mu y) w_y = c w + k \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Log[lambda*x]*D[w[x, y], x] + b*Log[mu*y]*D[w[x, y], y] == c*w[x,y]+k; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \frac {a c_1 \log (\lambda x) \exp \left (\int _1^x\frac {c K[1]-a}{a K[1] \log (\lambda K[1])}dK[1]\right )+b c_2 \log (\mu y) \exp \left (\int _1^y\frac {c K[1]-b}{b K[1] \log (\mu K[1])}dK[1]\right )-k}{c}\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*ln(lambda*x)*diff(w(x,y),x)+ b*ln(mu*y)*diff(w(x,y),y) =c*w(x,y)+k; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \frac {{\mathrm e}^{-\frac {c \,\operatorname {Ei}_{1}\left (-\ln \left (\lambda x \right )\right )}{a \lambda }} f_{1} \left (\frac {-a \,\operatorname {Ei}_{1}\left (-\ln \left (\mu y \right )\right ) \lambda +\operatorname {Ei}_{1}\left (-\ln \left (\lambda x \right )\right ) b \mu }{\lambda b \mu }\right ) c -k}{c}\]

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6.5.12.5 [1280] Problem 5

problem number 1280

Added April 5, 2019.

Problem Chapter 5.5.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a \ln ^n(\lambda x) w_x + b \ln ^m(\mu x) w_y = c \ln ^k(\nu x) w + p \ln ^s(\beta y)+q \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Log[lambda*x]^n*D[w[x, y], x] + b*Log[mu*x]^m*D[w[x, y], y] == c*Log[nu*x]^k*w[x,y]+p*Log[beta*y]^s+q; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \log ^{-n}(\lambda K[2]) \log ^k(\nu K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \log ^{-n}(\lambda K[2]) \log ^k(\nu K[2])}{a}dK[2]\right ) \log ^{-n}(\lambda K[3]) \left (p \log ^s\left (\beta \left (y-\int _1^x\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]+\int _1^{K[3]}\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]\right )\right )+q\right )}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*ln(lambda*x)^n*diff(w(x,y),x)+ b*ln(mu*x)^m*diff(w(x,y),y) = c*ln(nu*x)^k*w(x,y)+p*ln(beta*y)^s+q; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}{\mathrm e}^{-\frac {c \int \ln \left (\nu \textit {\_f} \right )^{k} \ln \left (\lambda \textit {\_f} \right )^{-n}d \textit {\_f}}{a}} \left (p {\ln \left (\frac {\beta \left (b \int \ln \left (\mu \textit {\_f} \right )^{m} \ln \left (\lambda \textit {\_f} \right )^{-n}d \textit {\_f} -b \int \ln \left (\mu x \right )^{m} \ln \left (\lambda x \right )^{-n}d x +y a \right )}{a}\right )}^{s}+q \right ) \ln \left (\lambda \textit {\_f} \right )^{-n}d \textit {\_f}}{a}+f_{1} \left (-\frac {b \int \ln \left (\mu x \right )^{m} \ln \left (\lambda x \right )^{-n}d x}{a}+y \right )\right ) {\mathrm e}^{\frac {c \int \ln \left (\nu x \right )^{k} \ln \left (\lambda x \right )^{-n}d x}{a}}\]

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6.5.12.6 [1281] Problem 6

problem number 1281

Added April 5, 2019.

Problem Chapter 5.5.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a \ln ^n(\lambda x) w_x + b \ln ^m(\mu x) w_y = c \ln ^k(\nu y) w + p \ln ^s(\beta x)+q \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Log[lambda*x]^n*D[w[x, y], x] + b*Log[mu*x]^m*D[w[x, y], y] == c*Log[nu*y]^k*w[x,y]+p*Log[beta*x]^s+q; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \log ^{-n}(\lambda K[2]) \log ^k\left (\nu \left (y-\int _1^x\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \log ^{-n}(\lambda K[2]) \log ^k\left (\nu \left (y-\int _1^x\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) \left (p \log ^s(\beta K[3])+q\right ) \log ^{-n}(\lambda K[3])}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*ln(lambda*x)^n*diff(w(x,y),x)+ b*ln(mu*x)^m*diff(w(x,y),y) = c*ln(nu*y)^k*w(x,y)+p*ln(beta*x)^s+q; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}{\mathrm e}^{-\frac {c \int {\ln \left (\frac {\nu \left (b \int \ln \left (\mu \textit {\_b} \right )^{m} \ln \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b} -b \int \ln \left (\mu x \right )^{m} \ln \left (\lambda x \right )^{-n}d x +y a \right )}{a}\right )}^{k} \ln \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b}}{a}} \left (p \ln \left (\beta \textit {\_b} \right )^{s}+q \right ) \ln \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b}}{a}+f_{1} \left (-\frac {b \int \ln \left (\mu x \right )^{m} \ln \left (\lambda x \right )^{-n}d x}{a}+y \right )\right ) {\mathrm e}^{\frac {c \int _{}^{x}{\ln \left (\frac {\nu \left (b \int \ln \left (\mu \textit {\_b} \right )^{m} \ln \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b} -b \int \ln \left (\mu x \right )^{m} \ln \left (\lambda x \right )^{-n}d x +y a \right )}{a}\right )}^{k} \ln \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b}}{a}}\]

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