6.3.25 8.1

6.3.25.1 [982] Problem 1

problem number 982

Added Feb. 11, 2019.

Problem Chapter 3.8.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = f(x) \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int \!{\frac {f \left ( x \right ) }{a}}\,{\rm d}x+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \]

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6.3.25.2 [983] Problem 2

problem number 983

Added Feb. 11, 2019.

Problem Chapter 3.8.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = y f(x) \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int ^{x}\!- \left ( \left ( x-{\it \_a} \right ) a-y \right ) f \left ( {\it \_a} \right ) {d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \]

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6.3.25.3 [984] Problem 3

problem number 984

Added Feb. 11, 2019.

Problem Chapter 3.8.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int ^{x}\! \left ( \left ( x-{\it \_a} \right ) a-y \right ) ^{2}f \left ( {\it \_a} \right ) + \left ( \left ( -x+{\it \_a} \right ) a+y \right ) g \left ( {\it \_a} \right ) +h \left ( {\it \_a} \right ) {d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \]

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6.3.25.4 [985] Problem 4

problem number 985

Added Feb. 11, 2019.

Problem Chapter 3.8.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int ^{x}\! \left ( \left ( -x+{\it \_a} \right ) a+y \right ) ^{k}f \left ( {\it \_a} \right ) {d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \]

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6.3.25.5 [986] Problem 5

problem number 986

Added Feb. 11, 2019.

Problem Chapter 3.8.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int ^{x}\!f \left ( {\it \_a} \right ) {{\rm e}^{- \left ( \left ( x-{\it \_a} \right ) a-y \right ) \lambda }}{d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \]

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6.3.25.6 [987] Problem 6

problem number 987

Added Feb. 11, 2019.

Problem Chapter 3.8.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int \!g \left ( x \right ) \,{\rm d}x+{\it \_F1} \left ( -\int \!f \left ( x \right ) {{\rm e}^{-ax}}\,{\rm d}x+y{{\rm e}^{-ax}} \right ) \]

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6.3.25.7 [988] Problem 7

problem number 988

Added Feb. 11, 2019.

Problem Chapter 3.8.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int ^{x}\! \left ( \left ( \int \!f \left ( {\it \_b} \right ) {{\rm e}^{-a{\it \_b}}}\,{\rm d}{\it \_b}-\int \!f \left ( x \right ) {{\rm e}^{-ax}}\,{\rm d}x+y{{\rm e}^{-ax}} \right ) {{\rm e}^{a{\it \_b}}} \right ) ^{k}g \left ( {\it \_b} \right ) {d{\it \_b}}+{\it \_F1} \left ( -\int \!f \left ( x \right ) {{\rm e}^{-ax}}\,{\rm d}x+y{{\rm e}^{-ax}} \right ) \]

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6.3.25.8 [989] Problem 8

problem number 989

Added Feb. 11, 2019.

Problem Chapter 3.8.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int \!g \left ( x \right ) \,{\rm d}x+{\it \_F1} \left ( {\frac {x \left ( k-1 \right ) {y}^{k}+y}{{y}^{k}}} \right ) \]

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6.3.25.9 [990] Problem 9

problem number 990

Added Feb. 11, 2019.

Problem Chapter 3.8.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\it \_F1} \left ( \left ( y+a \right ) {{\rm e}^{-x}} \right ) + \left ( \left ( -x+1 \right ) a+y \right ) b+cx\]

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6.3.25.10 [991] Problem 10

problem number 991

Added Feb. 11, 2019.

Problem Chapter 3.8.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int \!g \left ( x \right ) \,{\rm d}x+{\it \_F1} \left ( \left ( ax+a+y \right ) {{\rm e}^{-x}} \right ) \]

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6.3.25.11 [992] Problem 11

problem number 992

Added Feb. 11, 2019.

Problem Chapter 3.8.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int ^{x}\!{{\rm e}^{2\,\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it h2} \left ( {\it \_f} \right ) \left ( \int \!{\it g0} \left ( x \right ) {{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x \right ) ^{2}+ \left ( -2\,{\it h2} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}{{\rm e}^{2\,\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}y-2\,\int \!{\it g0} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}{{\rm e}^{2\,\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it h2} \left ( {\it \_f} \right ) -{{\rm e}^{\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it h1} \left ( {\it \_f} \right ) \right ) \int \!{\it g0} \left ( x \right ) {{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+ \left ( \int \!{\it g0} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f} \right ) ^{2}{{\rm e}^{2\,\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it h2} \left ( {\it \_f} \right ) + \left ( 2\,{\it h2} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}{{\rm e}^{2\,\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}y+{{\rm e}^{\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it h1} \left ( {\it \_f} \right ) \right ) \int \!{\it g0} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}+{\it h2} \left ( {\it \_f} \right ) \left ( {{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) ^{2}{{\rm e}^{2\,\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{y}^{2}+{{\rm e}^{\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}y{\it h1} \left ( {\it \_f} \right ) +{\it h0} \left ( {\it \_f} \right ) {d{\it \_f}}+{\it \_F1} \left ( -\int \!{\it g0} \left ( x \right ) {{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) \]

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6.3.25.12 [993] Problem 12

problem number 993

Added Feb. 11, 2019.

Problem Chapter 3.8.1.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int \!h \left ( x \right ) \,{\rm d}x+{\it \_F1} \left ( \left ( k-1 \right ) \int \!{{\rm e}^{ \left ( k-1 \right ) \int \!{\it g1} \left ( x \right ) \,{\rm d}x}}{\it g2} \left ( x \right ) \,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) \]

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6.3.25.13 [994] Problem 13

problem number 994

Added Feb. 11, 2019.

Problem Chapter 3.8.1.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

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

\[w \left ( x,y \right ) =\int \!{\frac {h \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x+{\it \_F1} \left ( {\frac {1}{\lambda } \left ( -\lambda \,\int \!{\frac {1}{f \left ( x \right ) }{{\rm e}^{\lambda \,\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x-{{\rm e}^{-\lambda \, \left ( y-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \right ) }} \right ) } \right ) \]

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6.3.25.14 [995] Problem 14

problem number 995

Added Feb. 11, 2019.

Problem Chapter 3.8.1.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int ^{x}\!{\frac {h \left ( {\it \_b} \right ) }{f \left ( {\it \_b} \right ) } \left ( \left ( \left ( 1+k \right ) \int \!{\frac {g \left ( {\it \_b} \right ) }{f \left ( {\it \_b} \right ) }}\,{\rm d}{\it \_b}+ \left ( -k-1 \right ) \int \!{\frac {g \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x+{y}^{k}y \right ) ^{ \left ( 1+k \right ) ^{-1}} \right ) ^{-k}}{d{\it \_b}}+{\it \_F1} \left ( \left ( -k-1 \right ) \int \!{\frac {g \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x+{y}^{k}y \right ) \]

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6.3.25.15 [996] Problem 15

problem number 996

Added Feb. 11, 2019.

Problem Chapter 3.8.1.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ y^k f(x) w_x + (y^{k+1} g_1(x) + g_0(x)) w_y = y^{3 k +1} h_2(x) + y^{2 k+1} h_1(x) + y^k h_0(x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  y^k*f[x]*D[w[x, y], x] + (y^(k + 1)*g1[x] + g0[x])*D[w[x, y], y] == y^(3*k + 1)*h2[x] + y^(2*k + 1)*h1[x] + y^k*h0[x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := y^k*f(x)*diff(w(x,y),x) +(y^(k+1)* g1(x) + g0(x))*diff(w(x,y),y) =   y^(3*k +1)*h2(x) + y^(2*k+1)*h1(x) + y^k*h0(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{f \left ( {\it \_f} \right ) } \left ( \left ( \left ( \left ( \left ( 1+k \right ) \int \!{\frac {{\it g0} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f} \left ( 1+k \right ) }}}\,{\rm d}{\it \_f}+ \left ( -k-1 \right ) \int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( 1+k \right ) }}}\,{\rm d}x+{y}^{1+k}{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( 1+k \right ) }} \right ) ^{ \left ( 1+k \right ) ^{-1}}{{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}} \right ) ^{k}{\it h1} \left ( {\it \_f} \right ) + \left ( \left ( \left ( 1+k \right ) \int \!{\frac {{\it g0} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f} \left ( 1+k \right ) }}}\,{\rm d}{\it \_f}+ \left ( -k-1 \right ) \int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( 1+k \right ) }}}\,{\rm d}x+{y}^{1+k}{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( 1+k \right ) }} \right ) ^{ \left ( 1+k \right ) ^{-1}}{{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}} \right ) ^{2\,k}{\it h2} \left ( {\it \_f} \right ) \right ) {{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}} \left ( \left ( 1+k \right ) \int \!{\frac {{\it g0} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f} \left ( 1+k \right ) }}}\,{\rm d}{\it \_f}+ \left ( -k-1 \right ) \int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( 1+k \right ) }}}\,{\rm d}x+{y}^{1+k}{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( 1+k \right ) }} \right ) ^{ \left ( 1+k \right ) ^{-1}}+{\it h0} \left ( {\it \_f} \right ) \right ) }{d{\it \_f}}+{\it \_F1} \left ( \left ( -k-1 \right ) \int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( 1+k \right ) }}}\,{\rm d}x+{y}^{1+k}{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( 1+k \right ) }} \right ) \]

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6.3.25.16 [997] Problem 16

problem number 997

Added Feb. 11, 2019.

Problem Chapter 3.8.1.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int \!{\frac {h \left ( x \right ) {{\rm e}^{-x\lambda }}}{f \left ( x \right ) }}\,{\rm d}x+{\it \_F1} \left ( -\int \!{\frac {g \left ( x \right ) {{\rm e}^{-x\lambda }}}{f \left ( x \right ) }}\,{\rm d}x+y \right ) \]

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