37 HFOPDE, chapter 1

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37.1 problem number 1

problem number 225

Added January 2, 2019.

Problem 1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x = f(x,y) \]

Mathematica ✓

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

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

Maple ✓

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

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

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37.2 problem number 2

problem number 226

Added January 2, 2019.

Problem 1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_y = f(x,y) \]

Mathematica ✓

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

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

Maple ✓

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

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

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37.3 problem number 3

problem number 227

Added January 2, 2019.

Problem 1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x = w f(x,y) \]

Mathematica ✓

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

\[ \left \{\left \{w(x,y)\to c_1(y) e^{\int _1^x f(K[1],y) \, dK[1]}\right \}\right \} \]

Maple ✓

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

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

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37.4 problem number 4

problem number 228

Added January 2, 2019.

Problem 1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_y = w f(x,y) \]

Mathematica ✓

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

\[ \left \{\left \{w(x,y)\to c_1(x) e^{\int _1^y f(x,K[1]) \, dK[1]}\right \}\right \} \]

Maple ✓

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

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

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37.5 problem number 5

problem number 229

Added January 2, 2019.

Problem 1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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37.6 problem number 6

problem number 230

Added January 2, 2019.

Problem 1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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