Added Feb. 25, 2019.
Problem Chapter 4.5.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = (c x^n+s \ln ^k(\lambda y)) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c*x^n + s*Log[gamma*y]^k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {s \log ^k(\gamma y) (-\log (\gamma y))^{-k} \operatorname {Gamma}(k+1,-\log (\gamma y))}{b \gamma }+\frac {c x^{n+1}}{a n+a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+b*diff(w(x,y),y) = (c*x^n+s*ln(gamma*y)^k)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\int _{}^{x}\frac {c \,\textit {\_a}^{n}+s \left (\ln \left (\frac {a y -\left (-\textit {\_a} +x \right ) b}{a}\right )+\ln \left (\gamma \right )\right )^{k}}{a}d \textit {\_a}}\]
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Added Feb. 25, 2019.
Problem Chapter 4.5.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a w_y = (b y^2+c x^n y+ s \ln ^k(\lambda x)) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*D[w[x, y], y] == (b*y^2 + c*x^n*y + s*Log[lambda*x]^k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1(y-a x) \exp \left (\frac {s \log ^k(\lambda x) (-\log (\lambda x))^{-k} \operatorname {Gamma}(k+1,-\log (\lambda x))}{\lambda }+\frac {1}{3} a^2 b x^3+a b x^2 (y-a x)+b x (y-a x)^2+\frac {c x^{n+1} ((n+2) y-a x)}{(n+1) (n+2)}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+a*diff(w(x,y),y) = (b*y^2+c*x^n*y+ s*ln(lambda*x)^k)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-a x +y \right ) {\mathrm e}^{\int _{}^{x}\left (a c \,\textit {\_a}^{n +1}-\left (a x -y \right ) c \,\textit {\_a}^{n}+s \ln \left (\textit {\_a} \lambda \right )^{k}+\left (\left (-\textit {\_a} +x \right ) a -y \right )^{2} b \right )d \textit {\_a}}\]
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Added March 9, 2019.
Problem Chapter 4.5.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a w_y = b \ln ^k(\lambda x) \ln ^n(\beta y) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*D[w[x, y], y] == b*Log[lambda*x]^k*Log[beta*y]^n*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1(y-a x) \exp \left (\int _1^xb \log ^k(\lambda K[1]) \log ^n(\beta (y+a (K[1]-x)))dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+a*diff(w(x,y),y) = b*ln(lambda*x)^k*ln(beta*y)^n*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-a x +y \right ) {\mathrm e}^{\int _{}^{x}b \ln \left (\textit {\_a} \lambda \right )^{k} \ln \left (-\left (\left (-\textit {\_a} +x \right ) a -y \right ) \beta \right )^{n}d \textit {\_a}}\]
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Added March 9, 2019.
Problem Chapter 4.5.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a y+b x^n) w_y = c \ln ^k(\lambda x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*y + b*x^n)*D[w[x, y], y] == c*Log[lambda*x]^k*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (b a^{-n-1} \operatorname {Gamma}(n+1,a x)+y e^{-a x}\right ) \exp \left (\frac {c (-\log (\lambda x))^{-k} \log ^k(\lambda x) \operatorname {Gamma}(k+1,-\log (\lambda x))}{\lambda }\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+(a*y+b*x^n)*diff(w(x,y),y) = c*ln(lambda*x)^k*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (-b \,x^{n} \left (a x \right )^{-\frac {n}{2}} \WhittakerM \left (\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, a x \right ) {\mathrm e}^{\frac {a x}{2}}+\left (n +1\right ) a y \right ) {\mathrm e}^{-a x}}{\left (n +1\right ) a}\right ) {\mathrm e}^{\int c \ln \left (\lambda x \right )^{k}d x}\]
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Added March 9, 2019.
Problem Chapter 4.5.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = x^k (n \ln x+ m \ln y) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == x^k*(n*Log[x] + m*Log[y])*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {b}{a}}\right ) \exp \left (\frac {x^k (a k m \log (y)+a k n \log (x)-a n-b m)}{a^2 k^2}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = x^k*(n*ln(x)+m*ln(y))*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = x^{\frac {n \,x^{k}}{a k}} \left (x^{\frac {b}{a}}\right )^{\frac {m \,x^{k}}{a k}} \left (y \,x^{-\frac {b}{a}}\right )^{\frac {m \,x^{k}}{a k}} \textit {\_F1} \left (y \,x^{-\frac {b}{a}}\right ) {\mathrm e}^{-\frac {\left (i \pi a k m \mathrm {csgn}\left (i y \right )^{3}-i \pi a k m \mathrm {csgn}\left (i y \right )^{2} \mathrm {csgn}\left (i x^{\frac {b}{a}}\right )-i \left (\mathrm {csgn}\left (i y \right )-\mathrm {csgn}\left (i x^{\frac {b}{a}}\right )\right ) \pi a k m \,\mathrm {csgn}\left (i y \right ) \mathrm {csgn}\left (i y \,x^{-\frac {b}{a}}\right )+2 a n +2 b m \right ) x^{k}}{2 a^{2} k^{2}}}\]
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Added March 9, 2019.
Problem Chapter 4.5.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^k w_x + b y^n w_y = (c \ln ^m(\lambda x)+s \ln ^t(\beta y)) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^k*D[w[x, y], x] + b*y^n*D[w[x, y], y] == (c*Log[lambda*x]^m + s*Log[beta*y]^t)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {b x^{1-k}}{a (k-1)}-\frac {y^{1-n}}{n-1}\right ) \exp \left (\int _1^x\frac {K[1]^{-k} \left (c \log ^m(\lambda K[1])+s \log ^t\left (\beta \left (\frac {a (k-1) x^k y^n K[1]^k}{a (k-1) x^k y K[1]^k-b (n-1) y^n \left (x K[1]^k-x^k K[1]\right )}\right )^{\frac {1}{n-1}}\right )\right )}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x^k*diff(w(x,y),x)+ b*y^n*diff(w(x,y),y) = (c*ln(lambda*x)^m+s*ln(beta*y)^t)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (k -1\right ) a \,y^{-n +1}-\left (n -1\right ) b \,x^{-k +1}}{\left (k -1\right ) a}\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (c \ln \left (\textit {\_a} \lambda \right )^{m}+s \ln \left (\beta \left (\frac {\left (k -1\right ) a \,y^{-n +1}+\left (n -1\right ) b \,\textit {\_a}^{-k +1}-\left (n -1\right ) b \,x^{-k +1}}{\left (k -1\right ) a}\right )^{-\frac {1}{n -1}}\right )^{t}\right ) \textit {\_a}^{-k}}{a}d \textit {\_a}}\]
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