6.2.27 8.4

6.2.27.1 [778] problem number 1

problem number 778

Added Feb. 7, 2019.

Problem 2.8.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x - \left (a y^2 \ln x -a x y (\ln x-1) f(x)+f(x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] - (a*y^2*Log[x] - a*x*y*(Log[x] - 1)*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)-(   a*y^2*ln(x) -a*x*y* (ln(x)-1)*f(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'));
 

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

____________________________________________________________________________________

6.2.27.2 [779] problem number 2

problem number 779

Added Feb. 7, 2019.

Problem 2.8.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left (f(x) y^2 -a x(\ln x) f(x) y+a \ln x+a \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^2 - a*x*Log[x]*f[x]*y + a*Log[x] + a)*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 -a*x*ln(x)*f(x)*y+a*ln(x)+a)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

sol=()

____________________________________________________________________________________

6.2.27.3 [780] problem number 3

problem number 780

Added Feb. 7, 2019.

Problem 2.8.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + \left (f(x) y^2 +a -a^2 (\ln x)^2 f(x) \right ) w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✗

restart; 
pde := x*diff(w(x,y),x)+( f(x)*y^2 +a -a^2* ln(x)^2 *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=()

____________________________________________________________________________________

6.2.27.4 [781] problem number 4

problem number 781

Added Feb. 7, 2019.

Problem 2.8.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + \left ((y+a \ln x)^2 f(x)-a \right ) w_y = 0 \]

Mathematica ✓

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

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

Maple ✓

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

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

____________________________________________________________________________________