6.3.15 5.3

6.3.15.1 [933] Problem 4

problem number 933

Added Feb. 11, 2019.

Problem Chapter 3.5.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + (a y+b x^n) w_y = c ln^k(\lambda x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*y + b*x^n)*D[w[x, y], y] == c*Log[lambda*x]^k; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \int _1^xc \log ^k(\lambda K[1])dK[1]+c_1\left (b a^{-n-1} \Gamma (n+1,a x)+y e^{-a x}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x) +  (a*y+b*x^n)*diff(w(x,y),y) =  c*ln(lambda*x)^k; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = c \int \ln \left (\lambda x \right )^{k}d x +f_{1} \left (\frac {\left (a x \right )^{-\frac {n}{2}} \left ({\mathrm e}^{-a x} y a \left (a x \right )^{\frac {n}{2}} \left (n +1\right )-{\mathrm e}^{-\frac {a x}{2}} b \,x^{n} \operatorname {WhittakerM}\left (\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, a x \right )\right )}{a \left (n +1\right )}\right )\]

Result has unresolved integrals

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6.3.15.2 [934] Problem 5

problem number 934

Added Feb. 11, 2019.

Problem Chapter 3.5.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x w_x + b y w_y = x^k ( n \ln x+ m \ln y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == x^k*(n*Log[x] + m*Log[y]); 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \frac {x^k (a k m \log (y)+a k n \log (x)-a n-b m)}{a^2 k^2}+c_1\left (y x^{-\frac {b}{a}}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := a*x*diff(w(x,y),x) +  b*y*diff(w(x,y),y) =  x^k*(n*ln(x)+m*ln(y)); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \frac {\left (2 \ln \left (x \right ) a k n -a \left (\left (i \left (\operatorname {csgn}\left (i x^{\frac {b}{a}}\right ) \operatorname {csgn}\left (i y \right ) \operatorname {csgn}\left (i y \,x^{-\frac {b}{a}}\right )-\operatorname {csgn}\left (i x^{\frac {b}{a}}\right )+\operatorname {csgn}\left (i y \right )-\operatorname {csgn}\left (i y \,x^{-\frac {b}{a}}\right )\right ) \pi -2 \ln \left (x^{\frac {b}{a}}\right )-2 \ln \left (y \,x^{-\frac {b}{a}}\right )\right ) m k +2 n \right )-2 m b \right ) x^{k}+2 f_{1} \left (y \,x^{-\frac {b}{a}}\right ) a^{2} k^{2}}{2 a^{2} k^{2}}\]

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6.3.15.3 [935] Problem 6

problem number 935

Added Feb. 11, 2019.

Problem Chapter 3.5.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x^k w_x + b y^n w_y = c \ln ^m(\lambda x)+s \ln ^l(\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^k*D[w[x, y], x] + b*y^n*D[w[x, y], y] == c*Log[lambda*x]^m + s*Log[beta*y]^l; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \int _1^x\frac {K[1]^{-k} \left (s \log ^l\left (\beta \left (\frac {a (k-1) x^k y^n K[1]^k}{a (k-1) x^k y K[1]^k-b (n-1) y^n \left (x K[1]^k-x^k K[1]\right )}\right )^{\frac {1}{n-1}}\right )+c \log ^m(\lambda K[1])\right )}{a}dK[1]+c_1\left (\frac {b x^{1-k}}{a (k-1)}-\frac {y^{1-n}}{n-1}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := a*x^k*diff(w(x,y),x) +  b*y^n*diff(w(x,y),y) =  c*ln(lambda*x)+s*ln(beta*y)^l; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \frac {f_{1} \left (\frac {-b \,x^{-k +1} \left (n -1\right )+y^{1-n} \left (k -1\right ) a}{\left (k -1\right ) a}\right ) a +\int _{}^{x}\left (c \ln \left (\lambda \textit {\_a} \right )+s {\ln \left (\beta \left (\frac {b \left (n -1\right ) \textit {\_a}^{-k +1}-b \,x^{-k +1} \left (n -1\right )+y^{1-n} \left (k -1\right ) a}{\left (k -1\right ) a}\right )^{-\frac {1}{n -1}}\right )}^{l}\right ) \textit {\_a}^{-k}d \textit {\_a}}{a}\]

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