Added January 29, 2019.
Problem 2.7.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a \arctan ^k(\lambda x)+b\right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*ArcTan[lambda*x]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\int _1^x\left (a \tan ^{-1}(\lambda K[1])^k+b\right )dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+( a*arctan(lambda*x)^k+b )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -bx+y-\int \!a \left ( \arctan \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x \right ) \]
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Added January 29, 2019.
Problem 2.7.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a \arctan ^k(\lambda y)+b\right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*ArcTan[lambda*y]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\frac {1}{a \tan ^{-1}(\lambda K[1])^k+b}dK[1]-x\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+( a*arctan(lambda*y)^k+b )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \arctan \left ( y\lambda \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \]
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Added January 29, 2019.
Problem 2.7.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + k \arctan ^n(a x+b y+c) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + k*ArcTan[a*x + b*y + c]^n*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)+k *arctan(a*x+b*y+c)^n*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac {ax+by}{b}}}\! \left ( k \left ( \arctan \left ( {\it \_a}\,b+c \right ) \right ) ^{n}b+a \right ) ^{-1}{d{\it \_a}}b+x \right ) \]
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Added January 29, 2019.
Problem 2.7.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + a \arctan ^k(\lambda x) \arctan ^n(\mu y) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*ArcTan[lambda*x]^k*ArcTan[mu*y]^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\tan ^{-1}(\mu K[1])^{-n}dK[1]-\int _1^xa \tan ^{-1}(\lambda K[2])^kdK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+a*arctan(lambda*x)^k*arctan(mu*y)^n*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( \arctan \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+\int \!{\frac { \left ( \arctan \left ( \mu \,y \right ) \right ) ^{-n}}{a}}\,{\rm d}y \right ) \]
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Added January 29, 2019.
Problem 2.7.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left (y^2 + \lambda (\arctan x)^n y -a^2 + a \lambda (\arctan x)^n \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + lambda*ArcTan[x]^n*y - a^2 + a*lambda*ArcTan[x]^n)*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 + lambda*arctan(x)^n*y -a^2 + a *lambda*arctan(x)^n )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( -a-y \right ) \int \!{{\rm e}^{-\int \!- \left ( \arctan \left ( x \right ) \right ) ^{n}\lambda +2\,a\,{\rm d}x}}\,{\rm d}x-{{\rm e}^{-\int \!- \left ( \arctan \left ( x \right ) \right ) ^{n}\lambda +2\,a\,{\rm d}x}}}{a+y}} \right ) \]
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Added January 29, 2019.
Problem 2.7.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left (y^2 + \lambda x (\arctan x)^n y + \lambda (\arctan x)^n \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + lambda*x*ArcTan[x]^n*y + lambda*ArcTan[x]^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\exp \left (-\int _1^x-\lambda \tan ^{-1}(K[5])^n K[5]dK[5]\right )}{x^2 y+x}-\int _1^x\frac {\exp \left (-\int _1^{K[6]}-\lambda \tan ^{-1}(K[5])^n K[5]dK[5]\right )}{K[6]^2}dK[6]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+(y^2 + lambda*x*arctan(x)^n*y + lambda*arctan(x)^n )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{xy+1} \left ( yx\int \!{{\rm e}^{\int \!{\frac { \left ( \arctan \left ( x \right ) \right ) ^{n}{x}^{2}\lambda -2}{x}}\,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\frac { \left ( \arctan \left ( x \right ) \right ) ^{n}{x}^{2}\lambda -2}{x}}\,{\rm d}x}}x+\int \!{{\rm e}^{\int \!{\frac { \left ( \arctan \left ( x \right ) \right ) ^{n}{x}^{2}\lambda -2}{x}}\,{\rm d}x}}\,{\rm d}x \right ) } \right ) \]
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Added Feb. 1, 2019.
Problem 2.7.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x - \left ( (k+1)x^k y^2 - \lambda (\arctan x)^n (x^{k+1}y-1)\right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] - ((k + 1)*x^k*y^2 - lambda*ArcTan[x]^n*(x^(k + 1)*y - 1))*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)-((k+1)*x^k*y^2 - lambda*arctan(x)^n*(x^(k+1)*y-1) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{{x}^{k+1}y-1} \left ( -{{\rm e}^{\int \!{\frac { \left ( \arctan \left ( x \right ) \right ) ^{n}{x}^{k+1}\lambda \,x-2\,k-2}{x}}\,{\rm d}x}}{x}^{k+1}+\int \!{\frac {{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \arctan \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x \left ( {x}^{k+1}y-1 \right ) \left ( k+1 \right ) \right ) } \right ) \]
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Added Feb. 1, 2019.
Problem 2.7.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( \lambda (\arctan x)^n +a y+ a b - b^2 \lambda (\arctan x)^n n\right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*ArcTan[x]^n + a*y + a*b - b^2*lambda*ArcTan[x]^n*n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} \left (\left (\lambda -b^2 \lambda n\right ) \tan ^{-1}(K[1])^n+a b\right )dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+(lambda* arctan(x)^n +a*y+ a*b - b^2*lambda*arctan(x)^n*n )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( \int \!-{{\rm e}^{-ax}} \left ( \left ( -{b}^{2}n+1 \right ) \lambda \, \left ( \arctan \left ( x \right ) \right ) ^{n}+ab \right ) \,{\rm d}x+y{{\rm e}^{-ax}} \right ) \]
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Added Feb. 1, 2019.
Problem 2.7.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( \lambda (arctan x)^n y^2 - b \lambda x^m (\arctan x)^n y+ b m x^{m-1} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*ArcTan[x]^n*y^2 - b*lambda*x^m*ArcTan[x]^n*y + b*m*x^(m - 1))*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)+(lambda*arctan(x)^n*y^2 - b*lambda*x^m*arctan(x)^n*y+ b*m*x^(m-1))*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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Added Feb. 1, 2019.
Problem 2.7.3.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( \lambda (arctan x)^n y^2 +b m x^{m-1} - \lambda b^2 x^{2 m}(\arctan x)^n \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*ArcTan[x]^n*y^2 + b*m*x^(m - 1) - lambda*b^2*x^(2*m)*ArcTan[x]^n)*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)+(lambda*arctan(x)^n*y^2 +b*m*x^(m-1) - lambda*b^2*x^(2*m)*arctan(x)^n)*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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Added Feb. 1, 2019.
Problem 2.7.3.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( \lambda (arctan x)^n (y-a x^m -b)^2 + a m x^{m-1} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*ArcTan[x]^n*(y - a*x^m - b)^2 + a*m*x^(m - 1))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\int _1^x\lambda \tan ^{-1}(K[2])^ndK[2]-\frac {1}{a x^m+b-y}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+(lambda*arctan(x)^n*(y-a*x^m -b)^2 + a*m*x^(m-1) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-1+ \left ( a{x}^{m}+b-y \right ) \int \!\lambda \, \left ( \arctan \left ( x \right ) \right ) ^{n}\,{\rm d}x}{a{x}^{m}+b-y}} \right ) \]
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Added Feb. 1, 2019.
Problem 2.7.3.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + \left ( \lambda (\arctan x)^n y^2+k y+ \lambda b^2 x^{2 k} (\arctan x)^n \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (lambda*ArcTan[x]^n*y^2 + k*y + lambda*b^2*x^(2*k)*ArcTan[x]^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y x^{-k}}{\sqrt {b^2}}\right )-\sqrt {b^2} \int _1^x\lambda \tan ^{-1}(K[1])^n K[1]^{k-1}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+(lambda*arctan(x)^n*y^2+k*y+lambda*b^2*x^(2*k)*arctan(x)^n )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( b\lambda \,\int \!{x}^{k-1} \left ( \arctan \left ( x \right ) \right ) ^{n}\,{\rm d}x-\arctan \left ( {\frac {y{x}^{-k}}{b}} \right ) \right ) \]
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