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'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\ln \left (\mathit {\_a} \lambda \right )^{k} \ln \left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta }{a}\right )^{n} {\mathrm e}^{-\frac {\mathit {\_a} c}{a}}}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]

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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'));
 

\[w \left (x , y\right ) = \left (\int \frac {s \ln \left (\beta x \right )^{n} {\mathrm e}^{-\frac {c \left (\int \ln \left (\lambda x \right )^{k}d x \right )}{a}}}{a}d x +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\int \frac {c \ln \left (\lambda x \right )^{k}}{a}d x}\]

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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]];
 

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = ( 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'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {k2} \mathit {s2} \ln \left (\frac {\left (a y -\left (-\mathit {\_b} +x \right ) b \right ) \beta 2}{a}\right )+\mathit {s1} \ln \left (\mathit {\_b} \beta 1 \right )^{\mathit {k1}}\right ) {\mathrm e}^{-\frac {\int \left (\mathit {c1} \ln \left (\mathit {\_b} \lambda 1 \right )^{\mathit {n1}}+\mathit {c2} \ln \left (\frac {\left (a y -\left (-\mathit {\_b} +x \right ) b \right ) \lambda 2}{a}\right )^{\mathit {n2}}\right )d \mathit {\_b}}{a}}}{a}d\mathit {\_b} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {\mathit {c1} \ln \left (\mathit {\_a} \lambda 1 \right )^{\mathit {n1}}+\mathit {c2} \ln \left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \lambda 2}{a}\right )^{\mathit {n2}}}{a}d\mathit {\_a}}\]

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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]];
 

Failed

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'));
 

\[w \left (x , y\right ) = \frac {c \mathit {\_F1} \left (\frac {-a \lambda \Ei \left (1, -\ln \left (\mu y \right )\right )+b \mu \Ei \left (1, -\ln \left (\lambda x \right )\right )}{b \lambda \mu }\right ) {\mathrm e}^{-\frac {c \Ei \left (1, -\ln \left (\lambda x \right )\right )}{a \lambda }}-k}{c}\]

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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'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {{\mathrm e}^{-\frac {c \left (\int \ln \left (\lambda \mathit {\_f} \right )^{-n} \ln \left (\nu \mathit {\_f} \right )^{k}d \mathit {\_f} \right )}{a}} \left (p \ln \left (\frac {\beta \left (b \left (\int \ln \left (\mu \mathit {\_f} \right )^{m} \ln \left (\lambda \mathit {\_f} \right )^{-n}d \mathit {\_f} \right )+\left (-\frac {b \left (\int \ln \left (\mu x \right )^{m} \ln \left (\lambda x \right )^{-n}d x \right )}{a}+y \right ) a \right )}{a}\right )^{s}+q \right ) \ln \left (\lambda \mathit {\_f} \right )^{-n}}{a}d\mathit {\_f} +\mathit {\_F1} \left (-\frac {b \left (\int \ln \left (\mu x \right )^{m} \ln \left (\lambda x \right )^{-n}d x \right )}{a}+y \right )\right ) {\mathrm e}^{\int \frac {\ln \left (\nu x \right )^{k} c \ln \left (\lambda x \right )^{-n}}{a}d x}\]

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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'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {{\mathrm e}^{-\frac {c \left (\int \ln \left (\frac {\nu \left (b \left (\int \ln \left (\mu \mathit {\_f} \right )^{m} \ln \left (\lambda \mathit {\_f} \right )^{-n}d \mathit {\_f} \right )+\left (-\frac {b \left (\int \ln \left (\mu x \right )^{m} \ln \left (\lambda x \right )^{-n}d x \right )}{a}+y \right ) a \right )}{a}\right )^{k} \ln \left (\lambda \mathit {\_f} \right )^{-n}d \mathit {\_f} \right )}{a}} \left (p \ln \left (\beta \mathit {\_f} \right )^{s}+q \right ) \ln \left (\lambda \mathit {\_f} \right )^{-n}}{a}d\mathit {\_f} +\mathit {\_F1} \left (-\frac {b \left (\int \ln \left (\mu x \right )^{m} \ln \left (\lambda x \right )^{-n}d x \right )}{a}+y \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {\ln \left (\nu \left (\int \frac {b \ln \left (\mu \mathit {\_b} \right )^{m} \ln \left (\lambda \mathit {\_b} \right )^{-n}}{a}d \mathit {\_b} -\frac {b \left (\int \ln \left (\mu x \right )^{m} \ln \left (\lambda x \right )^{-n}d x \right )}{a}+y \right )\right )^{k} c \ln \left (\lambda \mathit {\_b} \right )^{-n}}{a}d\mathit {\_b}}\]

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