6.4.24 8.1

6.4.24.1 [1160] Problem 1

problem number 1160

Added March 10, 2019.

Problem Chapter 4.8.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = f(x) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\int _1^x\frac {f(K[1])}{a}dK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {\int f \left (x \right )d x}{a}}\]

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6.4.24.2 [1161] Problem 2

problem number 1161

Added March 10, 2019.

Problem Chapter 4.8.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = f(x) y w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*D[w[x, y], y] == f[x]*y*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1(y-a x) \exp \left (\int _1^xf(K[1]) (y+a (K[1]-x))dK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (-a x +y \right ) {\mathrm e}^{\int _{}^{x}-\left (\left (x -\textit {\_a} \right ) a -y \right ) f \left (\textit {\_a} \right )d \textit {\_a}}\]

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6.4.24.3 [1162] Problem 3

problem number 1162

Added March 10, 2019.

Problem Chapter 4.8.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = (f(x) y^2+g(x) y+h(x)) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*D[w[x, y], y] == (f[x]*y^2 + g[x]*y + h[x])*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1(y-a x) \exp \left (\int _1^x\left (f(K[1]) (y+a (K[1]-x))^2+g(K[1]) (y+a (K[1]-x))+h(K[1])\right )dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+ a*diff(w(x,y),y) =(f(x)*y^2+g(x)*y+h(x))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-a x +y \right ) {\mathrm e}^{\int _{}^{x}\left (\left (\left (x -\textit {\_a} \right ) a -y \right )^{2} f \left (\textit {\_a} \right )+\left (\left (-x +\textit {\_a} \right ) a +y \right ) g \left (\textit {\_a} \right )+h \left (\textit {\_a} \right )\right )d \textit {\_a}}\]

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6.4.24.4 [1163] Problem 4

problem number 1163

Added March 10, 2019.

Problem Chapter 4.8.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = f(x) y^k w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*D[w[x, y], y] == f[x]*y^k*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1(y-a x) \exp \left (\int _1^xf(K[1]) (y+a (K[1]-x))^kdK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (-a x +y \right ) {\mathrm e}^{\int _{}^{x}f \left (\textit {\_a} \right ) \left (\left (-x +\textit {\_a} \right ) a +y \right )^{k}d \textit {\_a}}\]

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6.4.24.5 [1164] Problem 5

problem number 1164

Added March 10, 2019.

Problem Chapter 4.8.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = f(x) e^{\lambda y} w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*D[w[x, y], y] == f[x]*Exp[lambda*y]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1(y-a x) \exp \left (\int _1^xe^{\lambda (y+a (K[1]-x))} f(K[1])dK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (-a x +y \right ) {\mathrm e}^{\int _{}^{x}f \left (\textit {\_a} \right ) {\mathrm e}^{-\lambda \left (\left (x -\textit {\_a} \right ) a -y \right )}d \textit {\_a}}\]

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6.4.24.6 [1165] Problem 6

problem number 1165

Added March 10, 2019.

Problem Chapter 4.8.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y + f(x)) w_y = g(x) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*y + f[x])*D[w[x, y], y] == g[x]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^xg(K[2])dK[2]\right ) c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} f(K[1])dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+ (a*y+f(x))*diff(w(x,y),y) =g(x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left ({\mathrm e}^{-a x} y -\int f \left (x \right ) {\mathrm e}^{-a x}d x \right ) {\mathrm e}^{\int g \left (x \right )d x}\]

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6.4.24.7 [1166] Problem 7

problem number 1166

Added March 10, 2019.

Problem Chapter 4.8.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y + f(x)) w_y = g(x) y^k w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*y + f[x])*D[w[x, y], y] == g[x]*y^k*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} f(K[1])dK[1]\right ) \exp \left (\int _1^xg(K[2]) \left (e^{a K[2]} \left (e^{-a x} y-\int _1^xe^{-a K[1]} f(K[1])dK[1]+\int _1^{K[2]}e^{-a K[1]} f(K[1])dK[1]\right )\right ){}^kdK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+ (a*y+f(x))*diff(w(x,y),y) =g(x)*y^k*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left ({\mathrm e}^{-a x} y -\int f \left (x \right ) {\mathrm e}^{-a x}d x \right ) {\mathrm e}^{\int _{}^{x}g \left (\textit {\_b} \right ) {\left (\left (\int f \left (\textit {\_b} \right ) {\mathrm e}^{-a \textit {\_b}}d \textit {\_b} +{\mathrm e}^{-a x} y -\int f \left (x \right ) {\mathrm e}^{-a x}d x \right ) {\mathrm e}^{a \textit {\_b}}\right )}^{k}d \textit {\_b}}\]

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6.4.24.8 [1167] Problem 8

problem number 1167

Added March 10, 2019.

Problem Chapter 4.8.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + y^k w_y = g(x) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  f[x]*D[w[x, y], x] + y^k*D[w[x, y], y] == g[x]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {g(K[2])}{f(K[2])}dK[2]\right ) c_1\left (-\int _1^x\frac {1}{f(K[1])}dK[1]-\frac {y^{1-k}}{k-1}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  f(x)*diff(w(x,y),x)+ y^k*diff(w(x,y),y) =g(x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (y^{1-k}+\left (k -1\right ) \int \frac {1}{f \left (x \right )}d x \right ) {\mathrm e}^{\int \frac {g \left (x \right )}{f \left (x \right )}d x}\]

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6.4.24.9 [1168] Problem 9

problem number 1168

Added March 10, 2019.

Problem Chapter 4.8.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (y+a) w_y = (b y+c) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  f[x]*D[w[x, y], x] + (y + a)*D[w[x, y], y] == (b*y + c)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y \exp \left (-\int _1^x\frac {1}{f(K[1])}dK[1]\right )-\int _1^x\frac {a \exp \left (-\int _1^{K[2]}\frac {1}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right ) \exp \left (\int _1^x\frac {c+b \exp \left (\int _1^{K[3]}\frac {1}{f(K[1])}dK[1]\right ) \left (\exp \left (-\int _1^x\frac {1}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {a \exp \left (-\int _1^{K[2]}\frac {1}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {a \exp \left (-\int _1^{K[2]}\frac {1}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right )}{f(K[3])}dK[3]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\left (y +a \right ) {\mathrm e}^{-\int \frac {1}{f \left (x \right )}d x}\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (y +a \right ) b \,{\mathrm e}^{\int \frac {1}{f \left (\textit {\_b} \right )}d \textit {\_b} -\int \frac {1}{f \left (x \right )}d x}-a b +c}{f \left (\textit {\_b} \right )}d \textit {\_b}}\]

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6.4.24.10 [1169] Problem 10

problem number 1169

Added March 10, 2019.

Problem Chapter 4.8.1.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (y+a x) w_y = g(x) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  f[x]*D[w[x, y], x] + (y + a*x)*D[w[x, y], y] == g[x]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {g(K[3])}{f(K[3])}dK[3]\right ) c_1\left (y \exp \left (-\int _1^x\frac {1}{f(K[1])}dK[1]\right )-\int _1^x\frac {a \exp \left (-\int _1^{K[2]}\frac {1}{f(K[1])}dK[1]\right ) K[2]}{f(K[2])}dK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  f(x)*diff(w(x,y),x)+ (y+a*x)*diff(w(x,y),y) =g(x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-a \int \frac {x \,{\mathrm e}^{-\int \frac {1}{f \left (x \right )}d x}}{f \left (x \right )}d x +{\mathrm e}^{-\int \frac {1}{f \left (x \right )}d x} y \right ) {\mathrm e}^{\int \frac {g \left (x \right )}{f \left (x \right )}d x}\]

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6.4.24.11 [1170] Problem 11

problem number 1170

Added March 10, 2019.

Problem Chapter 4.8.1.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (g_1(x) y+g_0(x)) w_y = \left ( h_2(x) y^2+ h_1(x)y + h_0(x) \right ) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  f[x]*D[w[x, y], x] + (g1[x]*y + g0[x])*D[w[x, y], y] == (h2[x]*y^2 + h1[x]*y + h0[x])*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y \exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right ) \exp \left (\int _1^x\frac {\exp \left (2 \int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {h2}(K[3]) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right ){}^2+\exp \left (\int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {h1}(K[3]) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )+\text {h0}(K[3])}{f(K[3])}dK[3]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x))*diff(w(x,y),y) =(h2(x)*y^2+h1(x)*y+h0(x))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[\text {Expression too large to display}\]

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6.4.24.12 [1171] Problem 12

problem number 1171

Added March 10, 2019.

Problem Chapter 4.8.1.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (g_1(x) y+g_2(x) y^k) w_y = h(x) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  f[x]*D[w[x, y], x] + (g1[x]*y + g2[x]*y^k)*D[w[x, y], y] == h[x]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {h(K[3])}{f(K[3])}dK[3]\right ) c_1\left ((k-1) \int _1^x\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g2}(K[2])}{f(K[2])}dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  f(x)*diff(w(x,y),x)+ (g1(x)*y+g2(x)*y^k)*diff(w(x,y),y) =h(x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol := simplify(sol);
 

\[w \left (x , y\right ) = f_{1} \left (\left (k -1\right ) \int \frac {\operatorname {g2} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}\right ) {\mathrm e}^{\int \frac {h \left (x \right )}{f \left (x \right )}d x}\]

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6.4.24.13 [1172] Problem 13

problem number 1172

Added March 10, 2019.

Problem Chapter 4.8.1.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (g_1(x)+g_2(x) e^{\lambda y}) w_y = h(x) w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  f[x]*D[w[x, y], x] + (g1[x]*y + g2[x]*Exp[lambda*y])*D[w[x, y], y] == h[x]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  f(x)*diff(w(x,y),x)+ (g1(x)*y+g2(x)*exp(lambda*y))*diff(w(x,y),y) =h(x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.4.24.14 [1173] Problem 14

problem number 1173

Added March 10, 2019.

Problem Chapter 4.8.1.14, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) y^k w_x + g(x) w_y = h(x) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  f[x]*y^k*D[w[x, y], x] + g[x]*D[w[x, y], y] == h[x]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {y^{k+1}}{k+1}-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]\right ) \exp \left (\int _1^x\frac {h(K[2]) \left (\left (y^{k+1}-(k+1) \int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+(k+1) \int _1^{K[2]}\frac {g(K[1])}{f(K[1])}dK[1]\right ){}^{\frac {1}{k+1}}\right ){}^{-k}}{f(K[2])}dK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  f(x)*y^k*diff(w(x,y),x)+ g(x)*diff(w(x,y),y) =h(x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\left (-k -1\right ) \int \frac {g \left (x \right )}{f \left (x \right )}d x +y^{1+k}\right ) {\mathrm e}^{\int _{}^{x}\frac {h \left (\textit {\_b} \right ) {\left (\left (\int \frac {\left (1+k \right ) g \left (\textit {\_b} \right )}{f \left (\textit {\_b} \right )}d \textit {\_b} +\int \frac {\left (-k -1\right ) g \left (x \right )}{f \left (x \right )}d x +y^{1+k}\right )^{\frac {1}{1+k}}\right )}^{-k}}{f \left (\textit {\_b} \right )}d \textit {\_b}}\]

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6.4.24.15 [1174] Problem 15

problem number 1174

Added March 10, 2019.

Problem Chapter 4.8.1.15, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) e^{\lambda y} w_x + g(x) w_y = h(x) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  f[x]*Exp[lambda*y]*D[w[x, y], x] + g[x]*D[w[x, y], y] == h[x]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{\lambda y}}{\lambda }-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]\right ) \exp \left (\int _1^x\frac {h(K[2])}{f(K[2]) \left (-\lambda \int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+e^{\lambda y}+\lambda \int _1^{K[2]}\frac {g(K[1])}{f(K[1])}dK[1]\right )}dK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  f(x)*exp(lambda*y)*diff(w(x,y),x)+ g(x)*diff(w(x,y),y) =h(x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {{\mathrm e}^{\lambda y}-\int \frac {g \left (x \right )}{f \left (x \right )}d x \lambda }{\lambda }\right ) {\mathrm e}^{\int _{}^{x}\frac {h \left (\textit {\_b} \right )}{f \left (\textit {\_b} \right ) \left (\int \frac {g \left (\textit {\_b} \right )}{f \left (\textit {\_b} \right )}d \textit {\_b} \lambda -\int \frac {g \left (x \right )}{f \left (x \right )}d x \lambda +{\mathrm e}^{\lambda y}\right )}d \textit {\_b}}\]

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