6.2.29 8.6

6.2.29.1 [788] problem number 1

problem number 788

Added Feb. 7, 2019.

Problem 2.8.6.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2 -f(x) g(x) y+ g'(x)\right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^2 - f[x]*g[x]*y + Derivative[1][g][x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde := diff(w(x,y),x)+( f(x)*y^2 -f(x)*g(x)*y+ diff(g(x),x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

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6.2.29.2 [789] problem number 2

problem number 789

Added Feb. 7, 2019.

Problem 2.8.6.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x - \left ( f'(x) y^2 -f(x) g(x) y+ g(x)\right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] - (Derivative[1][f][x]*y^2 - f[x]*g[x]*y + g[x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)-( diff(f(x),x)*y^2 -f(x)*g(x)*y+ g(x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{f \left ( x \right ) y-1} \left ( f \left ( x \right ) y\int \!{\frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) {{\rm e}^{\int \!f \left ( x \right ) g \left ( x \right ) \,{\rm d}x}}}{ \left ( f \left ( x \right ) \right ) ^{2}}}\,{\rm d}x-f \left ( x \right ) {{\rm e}^{\int \!{\frac { \left ( f \left ( x \right ) \right ) ^{2}g \left ( x \right ) -2\,{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}-\int \!{\frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) {{\rm e}^{\int \!f \left ( x \right ) g \left ( x \right ) \,{\rm d}x}}}{ \left ( f \left ( x \right ) \right ) ^{2}}}\,{\rm d}x \right ) } \right ) \]

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6.2.29.3 [790] problem number 3

problem number 790

Added Feb. 7, 2019.

Problem 2.8.6.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( g(x)(y-f(x))^2 + f'(x)\right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (g[x]*(y - f[x])^2 + Derivative[1][f][x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := diff(w(x,y),x)+(g(x)*(y-f(x))^2 + diff(f(x),x) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.2.29.4 [791] problem number 4

problem number 791

Added Feb. 7, 2019.

Problem 2.8.6.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \frac {f'(x)}{g(x)} y^2 - \frac {g'(x)}{f(x)}\right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + ((Derivative[1][f][x]*y^2)/g[x] - Derivative[1][g][x]/f[x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+(diff(f(x),x)/g(x)* y^2 - diff(g(x),x)/f(x)  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{ \left ( f \left ( x \right ) y+g \left ( x \right ) \right ) f \left ( x \right ) } \left ( -1+ \left ( -y \left ( f \left ( x \right ) \right ) ^{2}-f \left ( x \right ) g \left ( x \right ) \right ) \int \!{\frac {{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{ \left ( f \left ( x \right ) \right ) ^{2}g \left ( x \right ) }}\,{\rm d}x \right ) } \right ) \]

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6.2.29.5 [792] problem number 5

problem number 792

Added Feb. 7, 2019.

Problem 2.8.6.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f^2(x) w_x + \left ( f'(x) y^2 -g(x) (y-f(x)) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  f[x]^2*D[w[x, y], x] + (Derivative[1][f][x]*y^2 - g[x]*(y - f[x]))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde := f(x)^2*diff(w(x,y),x)+(diff(f(x),x)*y^2 -g(x)*(y-f(x))  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

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6.2.29.6 [793] problem number 6

problem number 793

Added Feb. 7, 2019.

Problem 2.8.6.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2 - \frac {f''(x)}{f(x)} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 - Derivative[2][f][x]/f[x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+(y^2 - diff(f(x),x,x)/f(x) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-1+ \left ( -y \left ( f \left ( x \right ) \right ) ^{2}-f \left ( x \right ) {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) \int \! \left ( f \left ( x \right ) \right ) ^{-2}\,{\rm d}x}{ \left ( f \left ( x \right ) y+{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) f \left ( x \right ) }} \right ) \]

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6.2.29.7 [794] problem number 7

problem number 794

Added Feb. 7, 2019.

Problem 2.8.6.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ g(x) w_x + \left ( a f(x) g(x) y^3 + (b f(x) g^3(x) + g'(x)) y+ c f(x) g^4(x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  g[x]*D[w[x, y], x] + (a*f[x]*g[x]*y^3 + (b*f[x]*g[x]^3 + Derivative[1][g][x])*y + c*f[x]*g[x]^4)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  g(x)*diff(w(x,y),x)+(a*f(x)*g(x)*y^3 + (b*f(x)*g(x)^3 + diff(g(x),x))*y+ c*f(x)*g(x)^4)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {b}^{3}\sum _{{\it \_R}=\RootOf \left ( {c}^{2}a{{\it \_Z}}^{3}+{\it \_Z}\,{b}^{3}-{b}^{3} \right ) }{\frac {1}{3\,{{\it \_R}}^{2}a{c}^{2}+{b}^{3}}\ln \left ( {\frac {-{\it \_R}\,cg \left ( x \right ) -by}{cg \left ( x \right ) }} \right ) }-b\int \!f \left ( x \right ) \left ( g \left ( x \right ) \right ) ^{2}\,{\rm d}x \right ) \]

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6.2.29.8 [795] problem number 8

problem number 795

Added Feb. 7, 2019.

Problem 2.8.6.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^3+3 f(x) h(x) y^2+(g(x)+3 f(x) h^2(x)) y+ f(x) h^3(x) + g(x) h(x) - h'(x) \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^3 + 3*f[x]*h[x]*y^2 + (g[x] + 3*f[x]*h[x]^2)*y + f[x]*h[x]^3 + g[x]*h[x] - Derivative[1][h][x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := diff(w(x,y),x)+(f(x)*y^3+3*f(x)*h(x)*y^2+(g(x)+3*f(x)*h(x)^2)*y+ f(x)*h(x)^3 + g(x)* h(x) - diff(h(x),x) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {2\, \left ( y+h \left ( x \right ) \right ) ^{2}\int \!f \left ( x \right ) {{\rm e}^{2\,\int \!g \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{{\rm e}^{2\,\int \!g \left ( x \right ) \,{\rm d}x}}}{ \left ( y+h \left ( x \right ) \right ) ^{2}}} \right ) \]

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6.2.29.9 [796] problem number 9

problem number 796

Added Feb. 7, 2019.

Problem 2.8.6.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \frac {g'(x)}{f^2(x) (a g(x) + b)^3} y^3 + \frac {f'(x)}{f(x)} y + f(x) g'(x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + ((Derivative[1][g][x]*y^3)/(f[x]^2*(a*g[x] + b)^3) + (Derivative[1][f][x]*y)/f[x] + f[x]*Derivative[1][g][x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+(diff(g(x),x)/(f(x)^2 *(a*g(x)+b)^3)*y^3 + diff(f(x),x)/f(x) * y + f(x)*diff(g(x),x) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -{a}^{3}\sum _{{\it \_R}=\RootOf \left ( -{a}^{3}{\it \_Z}+{{\it \_Z}}^{3}+{a}^{3} \right ) }{\frac {1}{{a}^{3}-3\,{{\it \_R}}^{2}}\ln \left ( {\frac {-{\it \_R}\, \left ( ag \left ( x \right ) +b \right ) f \left ( x \right ) +ya}{ \left ( ag \left ( x \right ) +b \right ) f \left ( x \right ) }} \right ) }-\ln \left ( ag \left ( x \right ) +b \right ) \right ) \]

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6.2.29.10 [797] problem number 10

problem number 797

Added Feb. 7, 2019.

Problem 2.8.6.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( (y-f(x))(y-g(x)) \left (y- \frac {a f(x)+b g(x)}{a+b} \right ) h(x) + \frac {y-g(x)}{f(x)-g(x)} f'(x) + \frac {y-f(x)}{g(x)-f(x)} g'(x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + ((y - f[x])*(y - g[x])*(y - (a*f[x] + b*g[x])/(a + b))*h[x] + ((y - g[x])*Derivative[1][f][x])/(f[x] - g[x]) + ((y - f[x])*Derivative[1][g][x])/(g[x] - f[x]))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+((y-f(x))*(y-g(x))*(y- (a*f(x)+b*g(x))/(a+b))*h(x)+(y-g(x))/(f(x)-g(x))*diff(f(x),x)+ (y-f(x))/(g(x)-f(x))*diff(g(x),x) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( 1/3\,{\frac {{a}^{2}+ab+{b}^{2}}{ \left ( a+b \right ) ^{2}ba} \left ( - \left ( a+b \right ) ^{2}\ln \left ( 9\,{\frac { \left ( -af \left ( x \right ) -bg \left ( x \right ) + \left ( a+b \right ) y \right ) \left ( {a}^{2}+ab+{b}^{2} \right ) }{ \left ( f \left ( x \right ) -g \left ( x \right ) \right ) \left ( a-b \right ) }} \right ) +a \left ( a+b \right ) \ln \left ( 9\,{\frac { \left ( a+b \right ) \left ( {a}^{2}+ab+{b}^{2} \right ) \left ( y-f \left ( x \right ) \right ) }{ \left ( f \left ( x \right ) -g \left ( x \right ) \right ) \left ( a+2\,b \right ) }} \right ) +2\, \left ( \left ( a/2+b/2 \right ) \ln \left ( 9\,{\frac { \left ( a+b \right ) \left ( {a}^{2}+ab+{b}^{2} \right ) \left ( y-g \left ( x \right ) \right ) }{ \left ( f \left ( x \right ) -g \left ( x \right ) \right ) \left ( 2\,a+b \right ) }} \right ) +a \left ( \int \!h \left ( x \right ) f \left ( x \right ) g \left ( x \right ) \,{\rm d}x-1/2\,\int \! \left ( g \left ( x \right ) \right ) ^{2}h \left ( x \right ) \,{\rm d}x-1/2\,\int \!h \left ( x \right ) \left ( f \left ( x \right ) \right ) ^{2}\,{\rm d}x \right ) \right ) b \right ) } \right ) \]

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6.2.29.11 [798] problem number 11

problem number 798

Added Feb. 7, 2019.

Problem 2.8.6.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2 + g'(x) y+ a f(x) e^{2 g(x)} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^2 + Derivative[1][g][x]*y + a*f[x]*Exp[2*g[x]])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}, Assumptions -> a > 0], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y e^{-g(x)}}{\sqrt {a}}\right )-\sqrt {a} \int _1^xe^{g(K[1])} f(K[1])dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+(f(x)*y^2 + diff(g(x),x)*y+ a*f(x)*exp(2*g(x)) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) assuming a>0 ),output='realtime'));
 

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

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6.2.29.12 [799] problem number 12

problem number 799

Added Feb. 7, 2019.

Problem 2.8.6.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f'(x) y^2+ a e^{\lambda x} f(x) y+a e^{\lambda x} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (Derivative[1][f][x]*y^2 + a*Exp[lambda*x]*f[x]*y + a*Exp[lambda*x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+(diff(f(x),x)*y^2+ a*exp(lambda*x)* f(x)*y+a*exp(lambda*x) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))  ),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{f \left ( x \right ) y+1} \left ( -f \left ( x \right ) y\int \!{\frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) {{\rm e}^{a\int \!{{\rm e}^{\lambda \,x}}f \left ( x \right ) \,{\rm d}x}}}{ \left ( f \left ( x \right ) \right ) ^{2}}}\,{\rm d}x-f \left ( x \right ) {{\rm e}^{\int \!{\frac {{{\rm e}^{\lambda \,x}} \left ( f \left ( x \right ) \right ) ^{2}a-2\,{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}-\int \!{\frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) {{\rm e}^{a\int \!{{\rm e}^{\lambda \,x}}f \left ( x \right ) \,{\rm d}x}}}{ \left ( f \left ( x \right ) \right ) ^{2}}}\,{\rm d}x \right ) } \right ) \]

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