Added Feb. 1, 2019.
Problem 2.7.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a \arccot ^k(\lambda x)+b \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*ArcCot[lambda*x]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\int _1^x\left (\lambda \cot ^{-1}(\lambda K[1])^k+b\right )dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+(lambda*arccot(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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-b x +y -\left (\int \lambda \left (-\arctan \left (\lambda x \right )+\frac {\pi }{2}\right )^{k}d x \right )\right )\]
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Added Feb. 1, 2019.
Problem 2.7.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a \arccot ^k(\lambda y)+b \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*ArcCot[lambda*y]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\frac {1}{\lambda \cot ^{-1}(\lambda K[1])^k+b}dK[1]-x\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+(lambda*arccot(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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (x -\left (\int \frac {1}{\lambda \left (-\arctan \left (\lambda y \right )+\frac {\pi }{2}\right )^{k}+b}d y \right )\right )\]
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Added Feb. 1, 2019.
Problem 2.7.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + k \arccot ^n(a x+b y+c) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + k*ArcCot[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*arccot(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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-b \left (\int _{}^{\frac {a x +b y}{b}}\frac {1}{b k \left (-\arctan \left (\textit {\_a} b +c \right )+\frac {\pi }{2}\right )^{n}+a}d \textit {\_a} \right )+x \right )\]
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Added Feb. 1, 2019.
Problem 2.7.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + k \arccot ^k(\lambda x) \arccot ^n(\mu y) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*ArcCot[lambda*x]^k*ArcCot[lambda*y]^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\cot ^{-1}(\lambda K[1])^{-n}dK[1]-\int _1^xa \cot ^{-1}(\lambda K[2])^kdK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+a*arccot(lambda*x)^k*arccot(lambda*y)^n*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\left (\int \left (-\arctan \left (\lambda x \right )+\frac {\pi }{2}\right )^{k}d x \right )+\int \frac {\left (-\arctan \left (\lambda y \right )+\frac {\pi }{2}\right )^{-n}}{a}d y \right )\]
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Added Feb. 1, 2019.
Problem 2.7.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( y^2+ \lambda (\arccot x)^n y - a^2 +a \lambda (\arccot x)^n \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + lambda*ArcCot[x]^n*y - a^2 + a*lambda*ArcCot[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*arccot(x)^n*y - a^2 +a*lambda*arccot(x)^n)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (-a -y \right ) \left (\int {\mathrm e}^{-\left (\int \left (-\lambda \left (-\arctan \left (x \right )+\frac {\pi }{2}\right )^{n}+2 a \right )d x \right )}d x \right )-{\mathrm e}^{-\left (\int \left (-\lambda \left (-\arctan \left (x \right )+\frac {\pi }{2}\right )^{n}+2 a \right )d x \right )}}{a +y}\right )\]
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Added Feb. 1, 2019.
Problem 2.7.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( y^2+ \lambda x (\arccot x)^n y + \lambda (\arccot x)^n \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + lambda*x*ArcCot[x]^n*y + lambda*ArcCot[x]^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\exp \left (-\int _1^x-\lambda \cot ^{-1}(K[5])^n K[5]dK[5]\right )}{x^2 y+x}-\int _1^x\frac {\exp \left (-\int _1^{K[6]}-\lambda \cot ^{-1}(K[5])^n K[5]dK[5]\right )}{K[6]^2}dK[6]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+(y^2+lambda*x*arccot(x)^n*y +lambda*arccot(x)^n)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {x y \left (\int {\mathrm e}^{\int \frac {\lambda \,x^{2} \mathrm {arccot}\left (x \right )^{n}-2}{x}d x}d x \right )+x \,{\mathrm e}^{\int \frac {\lambda \,x^{2} \mathrm {arccot}\left (x \right )^{n}-2}{x}d x}+\int {\mathrm e}^{\int \frac {\lambda \,x^{2} \mathrm {arccot}\left (x \right )^{n}-2}{x}d x}d x}{y x +1}\right )\]
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Added Feb. 1, 2019.
Problem 2.7.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x - \left ( (k+1) x^k y^2- \lambda (\arccot x)^n (x^{k+1} y -1) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] - ((k + 1)*x^k*y^2 - lambda*ArcCot[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*arccot(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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-x^{k +1} {\mathrm e}^{\int \frac {\lambda x \,x^{k +1} \left (-\arctan \left (x \right )+\frac {\pi }{2}\right )^{n}-2 k -2}{x}d x}+\left (y \,x^{k +1}-1\right ) \left (k +1\right ) \left (\int \frac {x^{-k} {\mathrm e}^{\lambda \left (\int x^{k +1} \left (-\arctan \left (x \right )+\frac {\pi }{2}\right )^{n}d x \right )}}{x^{2}}d x \right )}{y \,x^{k +1}-1}\right )\]
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Added Feb. 1, 2019.
Problem 2.7.4.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( \lambda (\arccot x)^n y^2+a y + a b -b^2 \lambda (\arccot x)^n n \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*ArcCot[x]^n*y^2 + a*y + a*b - b^2*lambda*ArcCot[x]^n*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*arccot(x)^n*y^2+a*y + a*b -b^2*lambda*arccot(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'));
sol=()
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Added Feb. 1, 2019.
Problem 2.7.4.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( \lambda (\arccot x)^n y^2- b \lambda x^m(\arccot x)^n y+ b m x^{m-1} \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*ArcCot[x]^n*y^2 - b*lambda*x^m*ArcCot[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*arccot(x)^n*y^2- b*lambda*x^m*arccot(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 2.7.4.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( \lambda (\arccot x)^n y^2+ b m x^{m-1} - \lambda b^2 x^{2 m} (\arccot x^n) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*ArcCot[x]^n*y^2 + b*m*x^(m - 1) - lambda*b^2*x^(2*m)*ArcCot[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*arccot(x)^n*y^2+ b*m*x^(m-1) - lambda*b^2*x^(2*m)*arccot(x)^n )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
time expired
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Added Feb. 1, 2019.
Problem 2.7.4.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( \lambda (\arccot x)^n(y-a x^m-b)^2+a m x^{m-1} \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (lambda*ArcCot[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]];
\[\left \{\left \{w(x,y)\to c_1\left (\int _1^x\lambda \cot ^{-1}(K[2])^ndK[2]-\frac {1}{a x^m+b-y}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+( lambda*arccot(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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (a \,x^{m}+b -y \right ) \left (\int \lambda \mathrm {arccot}\left (x \right )^{n}d x \right )-1}{a \,x^{m}+b -y}\right )\]
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Added Feb. 1, 2019.
Problem 2.7.4.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + \left ( \lambda (\arccot x)^n y^2+ k y+ \lambda b^2 x^{2 k} (\arccot x)^n \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (lambda*ArcCot[x]^n*y^2 + k*y + lambda*b^2*x^(2*k)*ArcCot[x]^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y x^{-k}}{\sqrt {b^2}}\right )-\sqrt {b^2} \int _1^x\lambda \cot ^{-1}(K[1])^n K[1]^{k-1}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+( lambda*arccot(x)^n*y^2+ k*y+ lambda*b^2*x^(2*k)*arccot(x)^n )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (b \lambda \left (\int x^{k -1} \left (-\arctan \left (x \right )+\frac {\pi }{2}\right )^{n}d x \right )-\arctan \left (\frac {y \,x^{-k}}{b}\right )\right )\]
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