6.3.15 5.3

6.3.15.1 [934] Problem 4

problem number 934

Added Feb. 11, 2019.

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

Solve for $w(x,y)$

$$w_x+(ay+b{x}^n)w_y=cl{n}^k(\lambda x)$$

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]];
 

$$\left\{\{w(x,y)\to c_1(b{a}^{-n-1}\text{Gamma}(n+1,ax)+y{e}^{-ax})+\frac{c{\log }^k(\lambda x)(-\log (\lambda x){)}^{-k}\text{Gamma}(k+1,-\log (\lambda x))}{\lambda }\}\right\}$$

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'));
 

$$w(x,y)=\int c{(\ln \left(\lambda x\right))}^k\mathrm{d}x+\mathit{\_}\mathit{F}\mathit{1}\left(\frac{(-x^n{\left(ax\right)}^{-n/2}\mathrm{WhittakerM}(n/2,n/2+1/2,ax){\mathrm{e}}^{1/2ax}b+ay(n+1)){\mathrm{e}}^{-ax}}{a(n+1)}\right)$$ Result has unresolved integrals

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6.3.15.2 [935] Problem 5

problem number 935

Added Feb. 11, 2019.

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

Solve for $w(x,y)$

$$ax{w}_x+by{w}_y=x^k(n\ln x+m\ln y)$$

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]];
 

$$\left\{\{w(x,y)\to c_1\left(y{x}^{-\frac{b}{a}}\right)+\frac{x^k(akm\log (y)+akn\log (x)-an-bm)}{a^2{k}^2}\}\right\}$$

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'));
 

$$w(x,y)=1/2\frac{1}{k^2{a}^2}((2\ln \left(x\right)akn+(-m((i{\left(\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(iy\right)\right)}^3-i{\left(\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(iy\right)\right)}^2\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(iy{x}^{-\frac{b}{a}}\right)-i{\left(\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(iy\right)\right)}^2\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(i{x}^{\frac{b}{a}}\right)+i\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(iy\right)\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(iy{x}^{-\frac{b}{a}}\right)\mathit{c}\mathit{s}\mathit{g}\mathit{n}\left(i{x}^{\frac{b}{a}}\right))\pi -2\ln \left(x^{\frac{b}{a}}\right)-2\ln \left(y{x}^{-\frac{b}{a}}\right))k-2n)a-2bm)x^k+2\mathit{\_}\mathit{F}\mathit{1}\left(y{x}^{-\frac{b}{a}}\right)k^2{a}^2)$$

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6.3.15.3 [936] Problem 6

problem number 936

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]];
 

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

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'));
 

$$w(x,y)={\int }^x\frac{{\mathit{\_}\mathit{a}}^{-k}}{a}(c\ln \left(\mathit{\_}\mathit{a}\lambda \right)+s{(\ln \left(\beta {\left(\frac{-x^{1-k}b(n-1)+y^{-n+1}a(k-1)+{\mathit{\_}\mathit{a}}^{1-k}b(n-1)}{a(k-1)}\right)}^{-{(n-1)}^{-1}}\right))}^l)d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}\left(\frac{-x^{1-k}b(n-1)+y^{-n+1}a(k-1)}{a(k-1)}\right)$$

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