6.5.23 8.1
6.5.23.1 [1343] Problem 1
problem number 1343
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'));
\[w \left (x , y\right ) = \left (\frac {\int g \left (x \right ) {\mathrm e}^{-\frac {\int f \left (x \right )d x}{a}}d x}{a}+f_{1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {\int f \left (x \right )d x}{a}}\]
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6.5.23.2 [1344] Problem 2
problem number 1344
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'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}f \left (\textit {\_a} \right ) {\mathrm e}^{-\frac {\left (\left (c y +k \right ) a -\left (x -\frac {\textit {\_a}}{2}\right ) c b \right ) \textit {\_a}}{a^{2}}}d \textit {\_a}}{a}+f_{1} \left (\frac {y a -b x}{a}\right )\right ) {\mathrm e}^{\frac {x \left (\left (c y +k \right ) a -\frac {b c x}{2}\right )}{a^{2}}}\]
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6.5.23.3 [1345] Problem 3
problem number 1345
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'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}g \left (\textit {\_a} \right ) {\mathrm e}^{-\frac {\int \left (\left (-x +\textit {\_a} \right ) b +y a \right ) f \left (\textit {\_a} \right )d \textit {\_a}}{a^{2}}}d \textit {\_a}}{a}+f_{1} \left (\frac {y a -b x}{a}\right )\right ) {\mathrm e}^{\frac {\int _{}^{x}\left (\left (-x +\textit {\_a} \right ) b +y a \right ) f \left (\textit {\_a} \right )d \textit {\_a}}{a^{2}}}\]
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6.5.23.4 [1346] Problem 4
problem number 1346
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'));
\[w \left (x , y\right ) = \left (\frac {\int \frac {g \left (x \right ) {\mathrm e}^{-\frac {\int \frac {f \left (x \right )}{x}d x}{a}}}{x}d x}{a}+f_{1} \left (y \,x^{-\frac {b}{a}}\right )\right ) {\mathrm e}^{\frac {\int \frac {f \left (x \right )}{x}d x}{a}}\]
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6.5.23.5 [1347] Problem 5
problem number 1347
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'));
\[w \left (x , y\right ) = \left (\int \frac {g \left (x \right ) {\mathrm e}^{-c \int \frac {1}{f \left (x \right )}d x}}{f \left (x \right )}d x +f_{1} \left (\frac {{\mathrm e}^{-a \int \frac {1}{f \left (x \right )}d x} \left (a y +b \right )}{a}\right )\right ) {\mathrm e}^{c \int \frac {1}{f \left (x \right )}d x}\]
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6.5.23.6 [1348] Problem 6
problem number 1348
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'));
\[w \left (x , y\right ) = \left (\int \frac {p \left (x \right ) {\mathrm e}^{-\int \frac {h \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x +f_{1} \left (-\int \frac {g \left (x \right )}{f \left (x \right )}d x +y \right )\right ) {\mathrm e}^{\int \frac {h \left (x \right )}{f \left (x \right )}d x}\]
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6.5.23.7 [1349] Problem 7
problem number 1349
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'));
\[w \left (x , y\right ) = \left (\int \frac {\operatorname {h0} \left (x \right ) {\mathrm e}^{-\int \frac {\operatorname {h1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x +f_{1} \left ({\mathrm e}^{-\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x} y -\int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{-\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x \right )\right ) {\mathrm e}^{\int \frac {\operatorname {h1} \left (x \right )}{f \left (x \right )}d x}\]
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6.5.23.8 [1350] Problem 8
problem number 1350
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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {{\mathrm e}^{-\int \frac {\operatorname {h2} \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g}} \left (\operatorname {h1} \left (\textit {\_g} \right ) y \,{\mathrm e}^{\int \frac {\operatorname {g1} \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g} -\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}+{\mathrm e}^{\int \frac {\operatorname {g1} \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g}} \operatorname {h1} \left (\textit {\_g} \right ) \int \frac {\operatorname {g0} \left (\textit {\_g} \right ) {\mathrm e}^{-\int \frac {\operatorname {g1} \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g}}}{f \left (\textit {\_g} \right )}d \textit {\_g} -{\mathrm e}^{\int \frac {\operatorname {g1} \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g}} \operatorname {h1} \left (\textit {\_g} \right ) \int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{-\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x +\operatorname {h0} \left (\textit {\_g} \right )\right )}{f \left (\textit {\_g} \right )}d \textit {\_g} +f_{1} \left ({\mathrm e}^{-\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x} y -\int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{-\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x \right )\right ) {\mathrm e}^{\int \frac {\operatorname {h2} \left (x \right )}{f \left (x \right )}d x}\]
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6.5.23.9 [1351] Problem 9
problem number 1351
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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {{\mathrm e}^{-\int \frac {\operatorname {h2} \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g}} \left ({\left (\left (\left (1-k \right ) \int \frac {\operatorname {g0} \left (\textit {\_g} \right ) {\mathrm e}^{\int \frac {\operatorname {g1} \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g} \left (k -1\right )}}{f \left (\textit {\_g} \right )}d \textit {\_g} +\left (k -1\right ) \int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x \left (k -1\right )}}{f \left (x \right )}d x +y^{1-k} {\mathrm e}^{\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x \left (k -1\right )}\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \frac {\operatorname {g1} \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g}}\right )}^{n} \operatorname {h1} \left (\textit {\_g} \right )+\operatorname {h0} \left (\textit {\_g} \right )\right )}{f \left (\textit {\_g} \right )}d \textit {\_g} +f_{1} \left (\left (k -1\right ) \int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x \left (k -1\right )}}{f \left (x \right )}d x +y^{1-k} {\mathrm e}^{\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x \left (k -1\right )}\right )\right ) {\mathrm e}^{\int \frac {\operatorname {h2} \left (x \right )}{f \left (x \right )}d x}\]
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6.5.23.10 [1352] Problem 10
problem number 1352
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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {{\mathrm e}^{-\int \frac {\operatorname {h2} \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g}} \left (\operatorname {h1} \left (\textit {\_g} \right ) {\left (\frac {1}{\int -\frac {\lambda \,{\mathrm e}^{\int \frac {\lambda \operatorname {g1} \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g}} \operatorname {g0} \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g} +\int \frac {\lambda \,{\mathrm e}^{\int \frac {\lambda \operatorname {g1} \left (x \right )}{f \left (x \right )}d x} \operatorname {g0} \left (x \right )}{f \left (x \right )}d x +{\mathrm e}^{\lambda \left (\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x -y \right )}}\right )}^{\frac {\beta }{\lambda }} {\mathrm e}^{\beta \int \frac {\operatorname {g1} \left (\textit {\_g} \right )}{f \left (\textit {\_g} \right )}d \textit {\_g}}+\operatorname {h0} \left (\textit {\_g} \right )\right )}{f \left (\textit {\_g} \right )}d \textit {\_g} +f_{1} \left (\frac {-{\mathrm e}^{\lambda \left (\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x -y \right )}-\int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{\lambda \int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x \lambda }{\lambda }\right )\right ) {\mathrm e}^{\int \frac {\operatorname {h2} \left (x \right )}{f \left (x \right )}d x}\]
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6.5.23.11 [1353] Problem 11
problem number 1353
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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {h \left (\textit {\_b} \right ) {\left (\left (\int \frac {\left (1+k \right ) \operatorname {f2} \left (\textit {\_b} \right )}{\operatorname {f1} \left (\textit {\_b} \right )}d \textit {\_b} +\int \frac {\left (-k -1\right ) \operatorname {f2} \left (x \right )}{\operatorname {f1} \left (x \right )}d x +y^{1+k}\right )^{\frac {1}{1+k}}\right )}^{-k} {\mathrm e}^{-\int \frac {g \left (\textit {\_b} \right ) {\left (\left (\int \frac {\left (1+k \right ) \operatorname {f2} \left (\textit {\_b} \right )}{\operatorname {f1} \left (\textit {\_b} \right )}d \textit {\_b} +\int \frac {\left (-k -1\right ) \operatorname {f2} \left (x \right )}{\operatorname {f1} \left (x \right )}d x +y^{1+k}\right )^{\frac {1}{1+k}}\right )}^{-k}}{\operatorname {f1} \left (\textit {\_b} \right )}d \textit {\_b}}}{\operatorname {f1} \left (\textit {\_b} \right )}d \textit {\_b} +f_{1} \left (\left (-k -1\right ) \int \frac {\operatorname {f2} \left (x \right )}{\operatorname {f1} \left (x \right )}d x +y^{1+k}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {g \left (\textit {\_b} \right ) {\left (\left (\int \frac {\left (1+k \right ) \operatorname {f2} \left (\textit {\_b} \right )}{\operatorname {f1} \left (\textit {\_b} \right )}d \textit {\_b} +\int \frac {\left (-k -1\right ) \operatorname {f2} \left (x \right )}{\operatorname {f1} \left (x \right )}d x +y^{1+k}\right )^{\frac {1}{1+k}}\right )}^{-k}}{\operatorname {f1} \left (\textit {\_b} \right )}d \textit {\_b}}\]
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6.5.23.12 [1354] Problem 12
problem number 1354
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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {h \left (\textit {\_b} \right ) {\mathrm e}^{-\int \frac {g \left (\textit {\_b} \right )}{\operatorname {f1} \left (\textit {\_b} \right ) \left (\int \frac {\operatorname {f2} \left (\textit {\_b} \right )}{\operatorname {f1} \left (\textit {\_b} \right )}d \textit {\_b} \lambda -\int \frac {\operatorname {f2} \left (x \right )}{\operatorname {f1} \left (x \right )}d x \lambda +{\mathrm e}^{\lambda y}\right )}d \textit {\_b}}}{\operatorname {f1} \left (\textit {\_b} \right ) \left (\int \frac {\operatorname {f2} \left (\textit {\_b} \right )}{\operatorname {f1} \left (\textit {\_b} \right )}d \textit {\_b} \lambda -\int \frac {\operatorname {f2} \left (x \right )}{\operatorname {f1} \left (x \right )}d x \lambda +{\mathrm e}^{\lambda y}\right )}d \textit {\_b} +f_{1} \left (\frac {{\mathrm e}^{\lambda y}-\int \frac {\operatorname {f2} \left (x \right )}{\operatorname {f1} \left (x \right )}d x \lambda }{\lambda }\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {g \left (\textit {\_b} \right )}{\operatorname {f1} \left (\textit {\_b} \right ) \left (\int \frac {\operatorname {f2} \left (\textit {\_b} \right )}{\operatorname {f1} \left (\textit {\_b} \right )}d \textit {\_b} \lambda -\int \frac {\operatorname {f2} \left (x \right )}{\operatorname {f1} \left (x \right )}d x \lambda +{\mathrm e}^{\lambda y}\right )}d \textit {\_b}}\]
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