Added April 13, 2019.
Problem Chapter 5.8.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = f(x) w + g(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == f[x]*w[x,y]+g[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {f(K[1])}{a}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {f(K[1])}{a}dK[1]\right ) g(K[2])}{a}dK[2]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = f(x)*w(x,y)+g(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int \frac {{\mathrm e}^{-\frac {\int f \left (x \right )d x}{a}} g \left (x \right )}{a}d x +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\int \frac {f \left (x \right )}{a}d x}\]
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Added April 13, 2019.
Problem Chapter 5.8.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = (c y+k) w + f(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c*y+k)*w[x,y]+f[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x (2 a (c y+k)-b c x)}{2 a^2}} \left (\int _1^x\frac {\exp \left (-\frac {K[1] (2 a (k+c y)+b c (K[1]-2 x))}{2 a^2}\right ) f(K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c*y+k)*w(x,y)+f(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {{\mathrm e}^{-\frac {\left (-\left (-\frac {\mathit {\_a}}{2}+x \right ) b c +\left (c y +k \right ) a \right ) \mathit {\_a}}{a^{2}}} f \left (\mathit {\_a} \right )}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {\left (-\frac {b c x}{2}+\left (c y +k \right ) a \right ) x}{a^{2}}}\]
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Added April 13, 2019.
Problem Chapter 5.8.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = f(x) y w + g(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == f[x]*y*w[x,y]+g[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {f(K[1]) (a y+b (K[1]-x))}{a^2}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {f(K[1]) (a y+b (K[1]-x))}{a^2}dK[1]\right ) g(K[2])}{a}dK[2]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = f(x)*y*w(x,y)+g(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {{\mathrm e}^{-\frac {\int \left (a y +\left (\mathit {\_b} -x \right ) b \right ) f \left (\mathit {\_b} \right )d \mathit {\_b}}{a^{2}}} g \left (\mathit {\_b} \right )}{a}d\mathit {\_b} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) f \left (\mathit {\_a} \right )}{a^{2}}d\mathit {\_a}}\]
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Added April 13, 2019.
Problem Chapter 5.8.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = f(x) w + g(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == f[x]*w[x,y]+g[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {f(K[1])}{a K[1]}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {f(K[1])}{a K[1]}dK[1]\right ) g(K[2])}{a K[2]}dK[2]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = f(x)*w(x,y)+g(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int \frac {{\mathrm e}^{-\frac {\int \frac {f \left (x \right )}{x}d x}{a}} g \left (x \right )}{a x}d x +\mathit {\_F1} \left (y x^{-\frac {b}{a}}\right )\right ) {\mathrm e}^{\int \frac {f \left (x \right )}{a x}d x}\]
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Added April 13, 2019.
Problem Chapter 5.8.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (a y + b) w_y = c w + g(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (a+y+b)*D[w[x, y], y] == c*w[x,y]+g[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c}{f(K[3])}dK[3]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[4]}\frac {c}{f(K[3])}dK[3]\right ) g(K[4])}{f(K[4])}dK[4]+c_1\left (y \exp \left (-\int _1^x\frac {1}{f(K[1])}dK[1]\right )-\int _1^x\frac {(a+b) \exp \left (-\int _1^{K[2]}\frac {1}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ (a*y+b)*diff(w(x,y),y) = c*w(x,y)+g(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int \frac {{\mathrm e}^{-c \left (\int \frac {1}{f \left (x \right )}d x \right )} g \left (x \right )}{f \left (x \right )}d x +\mathit {\_F1} \left (\frac {\left (a y +b \right ) {\mathrm e}^{-a \left (\int \frac {1}{f \left (x \right )}d x \right )}}{a}\right )\right ) {\mathrm e}^{\int \frac {c}{f \left (x \right )}d x}\]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.8.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + g(x) w_y = h(x) w + p(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + g[x]*D[w[x, y], y] == h[x]*w[x,y]+p[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {h(K[2])}{f(K[2])}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {h(K[2])}{f(K[2])}dK[2]\right ) p(K[3])}{f(K[3])}dK[3]+c_1\left (y-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ g(x)*diff(w(x,y),y) = h(x)*w(x,y)+p(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int \frac {{\mathrm e}^{-\left (\int \frac {h \left (x \right )}{f \left (x \right )}d x \right )} p \left (x \right )}{f \left (x \right )}d x +\mathit {\_F1} \left (y -\left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )\right )\right ) {\mathrm e}^{\int \frac {h \left (x \right )}{f \left (x \right )}d x}\]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.8.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (g_1(x) y+ g_0(x)) w_y = h_1(x) w + h_0(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (g1[x]*y+g0[x])*D[w[x, y], y] == h1[x]*w[x,y]+h0[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {\text {h1}(K[3])}{f(K[3])}dK[3]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[4]}\frac {\text {h1}(K[3])}{f(K[3])}dK[3]\right ) \text {h0}(K[4])}{f(K[4])}dK[4]+c_1\left (y \exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x))*diff(w(x,y),y) = h1(x)*w(x,y)+h0(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int \frac {{\mathrm e}^{-\left (\int \frac {\mathit {h1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {h0} \left (x \right )}{f \left (x \right )}d x +\mathit {\_F1} \left (y \,{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}-\left (\int \frac {{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right )\right ) {\mathrm e}^{\int \frac {\mathit {h1} \left (x \right )}{f \left (x \right )}d x}\]
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Added April 13, 2019.
Problem Chapter 5.8.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (g_1(x) y+ g_0(x)) w_y = h_2(x) w + h_1(x) y + h0(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (g1[x]*y+g0[x])*D[w[x, y], y] == h2[x]*w[x,y]+h1[x]*y+h0[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {\text {h2}(K[3])}{f(K[3])}dK[3]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[4]}\frac {\text {h2}(K[3])}{f(K[3])}dK[3]\right ) \left (\text {h0}(K[4])+\exp \left (\int _1^{K[4]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {h1}(K[4]) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[4]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )}{f(K[4])}dK[4]+c_1\left (y \exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x))*diff(w(x,y),y) = h2(x)*w(x,y)+h1(x)*y+h0(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {{\mathrm e}^{-\left (\int \frac {\mathit {h2} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )} \mathit {h0} \left (\mathit {\_g} \right )+\left (y \,{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\int \frac {{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )} \mathit {g0} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} -\left (\int \frac {{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right ) {\mathrm e}^{\int \frac {\mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} -\left (\int \frac {\mathit {h2} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )} \mathit {h1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d\mathit {\_g} +\mathit {\_F1} \left (y \,{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}-\left (\int \frac {{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right )\right ) {\mathrm e}^{\int \frac {\mathit {h2} \left (x \right )}{f \left (x \right )}d x}\]
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Added April 13, 2019.
Problem Chapter 5.8.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (g_1(x) y+ g_0(x) y^k) w_y = h_2(x) w + h_1(x) y^n + h0(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (g1[x]*y+g0[x]*y^k)*D[w[x, y], y] == h2[x]*w[x,y]+h1[x]*y^n+h0[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {\text {h2}(K[3])}{f(K[3])}dK[3]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[4]}\frac {\text {h2}(K[3])}{f(K[3])}dK[3]\right ) \left (\text {h1}(K[4]) \left (\left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]-(k-1) \int _1^{K[4]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) (k-1) \int _1^x\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2] y^k-\exp \left (\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) (k-1) \int _1^{K[4]}\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2] y^k+\exp \left (k \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right ){}^n+\text {h0}(K[4])\right )}{f(K[4])}dK[4]+c_1\left ((k-1) \int _1^x\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x)*y^k)*diff(w(x,y),y) = h2(x)*w(x,y)+h1(x)*y^n+h0(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\left (\left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\left (-k +1\right ) \left (\int \frac {{\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )} \mathit {g0} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )+\left (k -1\right ) \left (\int \frac {{\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \frac {\mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g}}\right )^{n} \mathit {h1} \left (\mathit {\_g} \right )+\mathit {h0} \left (\mathit {\_g} \right )\right ) {\mathrm e}^{-\left (\int \frac {\mathit {h2} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )}}{f \left (\mathit {\_g} \right )}d\mathit {\_g} +\mathit {\_F1} \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\left (k -1\right ) \left (\int \frac {{\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right )\right ) {\mathrm e}^{\int \frac {\mathit {h2} \left (x \right )}{f \left (x \right )}d x}\]
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Added April 13, 2019.
Problem Chapter 5.8.1.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (g_1(x)+ g_0(x) e^{\lambda y}) w_y = h_2(x) w + h_1(x) e^{\beta y} + h0(x) \]
Mathematica ✗
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (g1[x]+g0[x]*Exp[lambda*y])*D[w[x, y], y] == h2[x]*w[x,y]+h1[x]*Exp[beta*y]+h0[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ (g1(x)+g0(x)*exp(lambda*y))*diff(w(x,y),y) = h2(x)*w(x,y)+h1(x)*exp(beta*y)+h0(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\frac {1}{\int \frac {\lambda \,{\mathrm e}^{\int \frac {\lambda \mathit {g1} \left (x \right )}{f \left (x \right )}d x} \mathit {g0} \left (x \right )}{f \left (x \right )}d x +\int -\frac {\lambda \,{\mathrm e}^{\int \frac {\lambda \mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g}} \mathit {g0} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} +{\mathrm e}^{\left (-y +\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right ) \lambda }}\right )^{\frac {\beta }{\lambda }} {\mathrm e}^{\beta \left (\int \frac {\mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )-\left (\int \frac {\mathit {h2} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )} \mathit {h1} \left (\mathit {\_g} \right )+{\mathrm e}^{-\left (\int \frac {\mathit {h2} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )} \mathit {h0} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d\mathit {\_g} +\mathit {\_F1} \left (\frac {-\lambda \left (\int \frac {{\mathrm e}^{\lambda \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )-{\mathrm e}^{\left (-y +\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right ) \lambda }}{\lambda }\right )\right ) {\mathrm e}^{\int \frac {\mathit {h2} \left (x \right )}{f \left (x \right )}d x}\]
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Added April 13, 2019.
Problem Chapter 5.8.1.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f_1(x) y^k w_x + f_2(x) w_y = g(x) w + h(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f1[x]*y^k*D[w[x, y], x] + f2[x]*D[w[x, y], y] == g[x]*w[x,y]+h[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {g(K[2]) \left (\left (y^{k+1}-(k+1) \int _1^x\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]+(k+1) \int _1^{K[2]}\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]\right ){}^{\frac {1}{k+1}}\right ){}^{-k}}{\text {f1}(K[2])}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {g(K[2]) \left (\left (y^{k+1}-(k+1) \int _1^x\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]+(k+1) \int _1^{K[2]}\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]\right ){}^{\frac {1}{k+1}}\right ){}^{-k}}{\text {f1}(K[2])}dK[2]\right ) h(K[3]) \left (\left (y^{k+1}-(k+1) \int _1^x\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]+(k+1) \int _1^{K[3]}\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]\right ){}^{\frac {1}{k+1}}\right ){}^{-k}}{\text {f1}(K[3])}dK[3]+c_1\left (\frac {y^{k+1}}{k+1}-\int _1^x\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := f1(x)*y^k*diff(w(x,y),x)+ f2(x)*diff(w(x,y),y) = g(x)*w(x,y)+h(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\left (y^{k +1}+\int \frac {\left (-k -1\right ) \mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x +\int \frac {\left (k +1\right ) \mathit {f2} \left (\mathit {\_f} \right )}{\mathit {f1} \left (\mathit {\_f} \right )}d \mathit {\_f} \right )^{\frac {1}{k +1}}\right )^{-k} {\mathrm e}^{-\left (\int \frac {\left (\left (y^{k +1}+\int \frac {\left (-k -1\right ) \mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x +\int \frac {\left (k +1\right ) \mathit {f2} \left (\mathit {\_f} \right )}{\mathit {f1} \left (\mathit {\_f} \right )}d \mathit {\_f} \right )^{\frac {1}{k +1}}\right )^{-k} g \left (\mathit {\_f} \right )}{\mathit {f1} \left (\mathit {\_f} \right )}d \mathit {\_f} \right )} h \left (\mathit {\_f} \right )}{\mathit {f1} \left (\mathit {\_f} \right )}d\mathit {\_f} +\mathit {\_F1} \left (y y^{k}+\left (-k -1\right ) \left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (\left (y^{k +1}+\int \frac {\left (-k -1\right ) \mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x +\int \frac {\left (k +1\right ) \mathit {f2} \left (\mathit {\_b} \right )}{\mathit {f1} \left (\mathit {\_b} \right )}d \mathit {\_b} \right )^{\frac {1}{k +1}}\right )^{-k} g \left (\mathit {\_b} \right )}{\mathit {f1} \left (\mathit {\_b} \right )}d\mathit {\_b}}\]
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Added April 13, 2019.
Problem Chapter 5.8.1.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f_1(x) e^{\lambda y} w_x + f_2(x) w_y = g(x) w + h(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f1[x]*Exp[lambda*y]*D[w[x, y], x] + f2[x]*D[w[x, y], y] == g[x]*w[x,y]+h[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {g(K[2])}{\text {f1}(K[2]) \left (-\lambda \int _1^x\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]+e^{\lambda y}+\lambda \int _1^{K[2]}\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]\right )}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {g(K[2])}{\text {f1}(K[2]) \left (-\lambda \int _1^x\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]+e^{\lambda y}+\lambda \int _1^{K[2]}\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]\right )}dK[2]\right ) h(K[3])}{\text {f1}(K[3]) \left (-\lambda \int _1^x\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]+e^{\lambda y}+\lambda \int _1^{K[3]}\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]\right )}dK[3]+c_1\left (\frac {e^{\lambda y}}{\lambda }-\int _1^x\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := f1(x)*exp(lambda*y)*diff(w(x,y),x)+ f2(x)*diff(w(x,y),y) = g(x)*w(x,y)+h(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {{\mathrm e}^{-\frac {\int \frac {g \left (\mathit {\_f} \right )}{\left (\int \frac {\mathit {f2} \left (\mathit {\_f} \right )}{\mathit {f1} \left (\mathit {\_f} \right )}d \mathit {\_f} +\frac {-\lambda \left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )+{\mathrm e}^{\lambda y}}{\lambda }\right ) \mathit {f1} \left (\mathit {\_f} \right )}d \mathit {\_f}}{\lambda }} h \left (\mathit {\_f} \right )}{\left (\lambda \left (\int \frac {\mathit {f2} \left (\mathit {\_f} \right )}{\mathit {f1} \left (\mathit {\_f} \right )}d \mathit {\_f} \right )-\lambda \left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )+{\mathrm e}^{\lambda y}\right ) \mathit {f1} \left (\mathit {\_f} \right )}d\mathit {\_f} +\mathit {\_F1} \left (\frac {-\lambda \left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )+{\mathrm e}^{\lambda y}}{\lambda }\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {g \left (\mathit {\_b} \right )}{\left (\lambda \left (\int \frac {\mathit {f2} \left (\mathit {\_b} \right )}{\mathit {f1} \left (\mathit {\_b} \right )}d \mathit {\_b} \right )-\lambda \left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )+{\mathrm e}^{\lambda y}\right ) \mathit {f1} \left (\mathit {\_b} \right )}d\mathit {\_b}}\]
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