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Added January 2, 2019.
Problem 2.2.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a y-b x}{a}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b'; pde:=a*diff(w(x,y),y)+b*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( x \right ) \] Solution missing \(a,b\) compared to book, but technically still correct.
Hand solution
\begin {equation} aw_{x}+bw_{y}=0\nonumber \end {equation}
The Lagrange-charpit equations are
\[ \frac {dx}{a}=\frac {dy}{b}=\frac {dw}{0}\]
The first pair of equations results in \(bdx=ady\) or \(bx=ay+C_{1}\). Hence \[ C_{1}=bx-ay \]
Since \(dw=0\) then \(w=C_{2}\). But \(C_{2}=F\left ( C_{1}\right ) \) where \(F\) is arbitrary function, therefore the solution is
\[ w\left ( x,y\right ) =F\left ( bx-ay\right ) \]
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Added January 2, 2019.
Problem 2.2.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + (b x+c) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c]; pde = a*D[w[x, y], x] + (b*x + c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {2 a y-b x^2-2 c x}{2 a}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c'; pde:=a*diff(w(x,y),x)+(b*x+c)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac {-b{x}^{2}+2\,ya-2\,cx}{a}} \right ) \]
Hand solution
Solve \(aw_{x}+\left ( bx+c\right ) w_{y}=0\). The Lagrange-charpit equations are
\[ \frac {dx}{a}=\frac {dy}{\left ( bx+c\right ) }=\frac {dw}{0}\]
The first pair of equations gives \(\frac {\left ( bx+c\right ) }{a}dx=dy\). Integrating results in
\begin {align*} \frac {1}{a}\left ( \frac {bx^{2}}{2}+cx\right ) & =y+C_{1}\\ C_{1} & =\frac {1}{a}\left ( \frac {bx^{2}}{2}+cx\right ) -y \end {align*}
Since \(dw=0\) then \(w=C_{2}\). But \(C_{2}=F\left ( C_{1}\right ) \). Where \(F\) is arbitray function. Therefore
\[ w\left ( x,y\right ) =F\left ( \frac {bx^{2}}{2a}+\frac {c}{a}x-y\right ) \]
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Added January 2, 2019.
Problem 2.2.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a x+b y+c) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c]; pde = D[w[x, y], x] + (a*x + b*y + c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {e^{-b x} \left (a b x+a+b^2 y+b c\right )}{b^2}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c'; pde:=diff(w(x,y),x)+(a*x+b*y+c)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( axb+y{b}^{2}+bc+a \right ) {{\rm e}^{-bx}}}{{b}^{2}}} \right ) \]
Hand solution
Solve\begin {equation} w_{x}+\left ( ax+by+c\right ) w_{y}=0\tag {1} \end {equation} The Lagrange-charpit equations are \[ dx=\frac {dy}{\left ( ax+by+c\right ) }=\frac {dw}{0}\] The first pair of equations gives \begin {align*} \frac {dy}{dx} & =ax+by+c\\ \frac {dy}{dx}-by & =ax+c \end {align*}
This is linear. Integrating factor is \(I=e^{-bx}\). Hence the above becomes\begin {align*} \frac {d}{dx}\left ( ye^{-bx}\right ) & =\left ( ax+c\right ) e^{-bx}\\ ye^{-bx} & =a\int xe^{-bx}+c\int e^{-bx}+C_{1}\\ ye^{-bx} & =a\left ( -\frac {\left ( 1+bx\right ) e^{-bx}}{b^{2}}\right ) -c\frac {e^{-bx}}{b}+C_{1}\\ y & =-a\frac {\left ( 1+bx\right ) }{b^{2}}-\frac {c}{b}-C_{1}e^{bx}\\ C_{1} & =-\left ( y+\frac {a}{b^{2}}\left ( 1+bx\right ) +\frac {c}{b}\right ) e^{-bx} \end {align*}
Since \(dw=0\) then \(w=C_{2}\). But \(C_{2}=F\left ( C_{1}\right ) \). Therefore\begin {align*} w\left ( x,y\right ) & =F\left ( -\left ( y+\frac {a}{b^{2}}\left ( 1+bx\right ) +\frac {c}{b}\right ) e^{-bx}\right ) \\ & =F\left ( \left ( y+\frac {a}{b^{2}}\left ( 1+bx\right ) +\frac {c}{b}\right ) e^{-bx}\right ) \end {align*}
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Added January 2, 2019.
Problem 2.2.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {b}{a}}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c'; pde:=a*x*diff(w(x,y),x)+b*y*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \]
Hand solution
Solve\begin {equation} axw_{x}+byw_{y}=0\tag {1} \end {equation} The Lagrange-charpit equations are \[ \frac {dx}{ax}=\frac {dy}{by}=\frac {dw}{0}\] The first pair of equations gives \begin {align*} \frac {b}{a}\frac {dx}{x} & =\frac {dy}{y}\\ \frac {b}{a}\ln x & =\ln y+C_{1}\\ x^{\frac {b}{a}} & =C_{1}y\\ C_{1} & =\frac {x^{\frac {b}{a}}}{y} \end {align*}
Since \(dw=0\) then \(w=C_{2}\). But \(C_{2}=F\left ( C_{1}\right ) \). Therefore\[ w\left ( x,y\right ) =F\left ( \frac {x^{\frac {b}{a}}}{y}\right ) \]
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Added January 2, 2019.
Problem 2.2.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y w_x + b x w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b]; pde = a*y*D[w[x, y], x] + b*x*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a y^2-b x^2}{2 a}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b'; pde:=a*y*diff(w(x,y),x)+b*x*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{y}^{2}a-b{x}^{2}}{a}} \right ) \]
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Added January 2, 2019.
Problem 2.2.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ y w_x + (y+a) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a]; pde = y*D[w[x, y], x] + (y + a)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (a \left (-\log \left (a \left (\frac {y e^{-\frac {y}{a}-1}}{a}+e^{-\frac {y}{a}-1}\right )\right )\right )-a-x\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a'; pde:=y*diff(w(x,y),x)+(y+a)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -a\ln \left ( y+a \right ) +y-x \right ) \]
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Added January 2, 2019.
Problem 2.2.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a y + b x + c ) w_x - (b y + k x +s) w_y =0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c, s]; pde = (a*y + b*x + c)*D[w[x, y], x] - (b*y + k*x + s)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a y^2+2 b x y+2 c y+k x^2+2 s x}{a}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c';s:='s'; pde:=(a*y+b*x+c)*diff(w(x,y),x)-(b*y+k*x+s)*diff(w(x,y),y)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt {{a}^{3}{k}^{2}{y}^{2}-2\,{a}^{2}{b}^{2}k{y}^{2}+2\,{a}^{2}b{k}^{2}xy+{a}^{2}{k}^{3}{x}^{2}+a{b}^{4}{y}^{2}-4\,a{b}^{3}kxy-2\,a{b}^{2}{k}^{2}{x}^{2}+2\,{b}^{5}xy+{b}^{4}k{x}^{2}+2\,{a}^{2}c{k}^{2}y+2\,{a}^{2}{k}^{2}sx-4\,a{b}^{2}cky-4\,a{b}^{2}ksx+2\,{b}^{4}cy+2\,{b}^{4}sx+{a}^{2}k{s}^{2}-a{b}^{2}{s}^{2}-2\,abcks+a{c}^{2}{k}^{2}+2\,{b}^{3}cs-{c}^{2}k{b}^{2}}}} \right ) \] But Mathematica solution is simpler, both verified correct
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Added January 2, 2019.
Problem 2.2.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a_1 x + b_1 y + c_1 ) w_x + (a_2 x + b_2 y +c_2) w_y =0 \]
Mathematica ✗
ClearAll[w, x, y, a1, a2, b1, b2, c1, c2]; pde = (a1*x + b1*y + c1)*D[w[x, y], x] + (a1*x + b2*y + c2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a1:='a1';b1:='b1';c1:='c1';a2:='a2';b2:='b2';c2:='c2'; pde:=(a1*x+b1*y+c1)*diff(w(x,y),x)+(a2*x+b2*y+c2)*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 ( -1/2\,{\frac {1}{\sqrt {-{{\it a1}}^{2}+2\,{\it a1}\,{\it b2}-4\,{\it a2}\,{\it b1}-{{\it b2}}^{2}}} \left ( \sqrt {-{{\it a1}}^{2}+2\,{\it a1}\,{\it b2}-4\,{\it a2}\,{\it b1}-{{\it b2}}^{2}}\ln \left ( y{{\it a1}}^{3}{{\it b2}}^{2}x-2\,y{{\it a1}}^{2}{\it a2}\,{\it b1}\,{\it b2}\,x-{{\it a1}}^{2}{\it a2}\,{{\it b2}}^{2}{x}^{2}+{{\it a1}}^{2}{\it b1}\,{{\it b2}}^{2}{y}^{2}-y{{\it a1}}^{2}{{\it b2}}^{3}x+y{\it a1}\,{{\it a2}}^{2}{{\it b1}}^{2}x+2\,{\it a1}\,{{\it a2}}^{2}{\it b1}\,{\it b2}\,{x}^{2}-2\,{\it a1}\,{\it a2}\,{{\it b1}}^{2}{\it b2}\,{y}^{2}+2\,y{\it a1}\,{\it a2}\,{\it b1}\,{{\it b2}}^{2}x-{{\it a2}}^{3}{{\it b1}}^{2}{x}^{2}+{{\it a2}}^{2}{{\it b1}}^{3}{y}^{2}-y{{\it a2}}^{2}{{\it b1}}^{2}{\it b2}\,x+{{\it a1}}^{3}{\it b2}\,{\it c2}\,x-{{\it a1}}^{2}{\it a2}\,{\it b1}\,{\it c2}\,x-{{\it a1}}^{2}{\it a2}\,{\it b2}\,{\it c1}\,x+{{\it a1}}^{2}{\it b1}\,{\it b2}\,{\it c2}\,y+{{\it a1}}^{2}{{\it b2}}^{2}{\it c1}\,y-{{\it a1}}^{2}{{\it b2}}^{2}{\it c2}\,x+{\it a1}\,{{\it a2}}^{2}{\it b1}\,{\it c1}\,x-{\it a1}\,{\it a2}\,{{\it b1}}^{2}{\it c2}\,y-3\,{\it a1}\,{\it a2}\,{\it b1}\,{\it b2}\,{\it c1}\,y+3\,{\it a1}\,{\it a2}\,{\it b1}\,{\it b2}\,{\it c2}\,x-{\it a1}\,{\it a2}\,{{\it b2}}^{2}{\it c1}\,x+{\it a1}\,{\it b1}\,{{\it b2}}^{2}{\it c2}\,y-{\it a1}\,{{\it b2}}^{3}{\it c1}\,y+2\,{{\it a2}}^{2}{{\it b1}}^{2}{\it c1}\,y-2\,{{\it a2}}^{2}{{\it b1}}^{2}{\it c2}\,x+{{\it a2}}^{2}{\it b1}\,{\it b2}\,{\it c1}\,x-{\it a2}\,{{\it b1}}^{2}{\it b2}\,{\it c2}\,y+{\it a2}\,{\it b1}\,{{\it b2}}^{2}{\it c1}\,y+{{\it a1}}^{2}{\it b2}\,{\it c1}\,{\it c2}-{\it a1}\,{\it a2}\,{\it b1}\,{\it c1}\,{\it c2}-{\it a1}\,{\it a2}\,{\it b2}\,{{\it c1}}^{2}+{\it a1}\,{\it b1}\,{\it b2}\,{{\it c2}}^{2}-{\it a1}\,{{\it b2}}^{2}{\it c1}\,{\it c2}+{{\it a2}}^{2}{\it b1}\,{{\it c1}}^{2}-{\it a2}\,{{\it b1}}^{2}{{\it c2}}^{2}+{\it a2}\,{\it b1}\,{\it b2}\,{\it c1}\,{\it c2} \right ) +2\,\arctan \left ( {\frac {{{\it a1}}^{2}{\it b2}\,x-{\it a1}\,{\it a2}\,{\it b1}\,x+2\,{\it a1}\,{\it b1}\,{\it b2}\,y-{\it a1}\,{{\it b2}}^{2}x-2\,{\it a2}\,{{\it b1}}^{2}y+{\it a2}\,{\it b1}\,{\it b2}\,x+{\it a1}\,{\it c2}\,{\it b1}+{\it b2}\,{\it c1}\,{\it a1}-2\,{\it a2}\,{\it c1}\,{\it b1}+{\it b1}\,{\it c2}\,{\it b2}-{{\it b2}}^{2}{\it c1}}{\sqrt {-{{\it a1}}^{2}+2\,{\it a1}\,{\it b2}-4\,{\it a2}\,{\it b1}-{{\it b2}}^{2}} \left ( {\it a1}\,{\it b2}\,x-{\it a2}\,{\it b1}\,x-{\it b1}\,{\it c2}+{\it b2}\,{\it c1} \right ) }} \right ) {\it a1}+2\,\arctan \left ( {\frac {{{\it a1}}^{2}{\it b2}\,x-{\it a1}\,{\it a2}\,{\it b1}\,x+2\,{\it a1}\,{\it b1}\,{\it b2}\,y-{\it a1}\,{{\it b2}}^{2}x-2\,{\it a2}\,{{\it b1}}^{2}y+{\it a2}\,{\it b1}\,{\it b2}\,x+{\it a1}\,{\it c2}\,{\it b1}+{\it b2}\,{\it c1}\,{\it a1}-2\,{\it a2}\,{\it c1}\,{\it b1}+{\it b1}\,{\it c2}\,{\it b2}-{{\it b2}}^{2}{\it c1}}{\sqrt {-{{\it a1}}^{2}+2\,{\it a1}\,{\it b2}-4\,{\it a2}\,{\it b1}-{{\it b2}}^{2}} \left ( {\it a1}\,{\it b2}\,x-{\it a2}\,{\it b1}\,x-{\it b1}\,{\it c2}+{\it b2}\,{\it c1} \right ) }} \right ) {\it b2} \right ) } \right ) \]