share/contrib/diffequations/tests/rtest_ode1_riccati.mac

   1 (load("contrib_ode"),0);
   2 0$
   3
   4 /* Test cases for ode1_riccati.mac functions */
   5
   6 /* Test cases with n=2*a   ode1_riccati_special(a,b,c,n,y,x)*/
   7 eq:x*'diff(y,x)-a*y+b*y^2=c*x^n;
   8 x*'diff(y,x)-a*y+b*y^2=c*x^n;
   9
  10 /* Case a.i.1 n=2*a, b>0, c>0 */
  11 (print("Special Riccati equation: Case a.i.1: n=2*a, b>0, c>0"),done);
  12 done;
  13 ans:ode1_riccati_special(a:1,b:1,c:1,n:2,y,x);
  14 y=x*tanh(x+%c);
  15 ode_check(eq,ans);
  16 0;
  17 ans:ode1_riccati(eqn:x*'diff(y,x)-y+y^2=x^2,y,x);
  18 y=x*tanh(x+%c);
  19 ode_check(eqn,ans);
  20 0;
  21 ans:ode1_riccati_special(a:2,b:2,c:3,n:4,y,x);
  22 y=sqrt(3)*x^2*tanh(sqrt(6)*x^2/2+%c)/sqrt(2);
  23 radcan(ode_check(eq,ans));
  24 0;
  25 ans:ode1_riccati_special(a:3,b:2,c:3,n:6,y,x);
  26 y=sqrt(3)*x^3*tanh(sqrt(6)*x^3/3+%c)/sqrt(2);
  27 radcan(ode_check(eq,ans));
  28 0;
  29 ans:ode1_riccati_special(a:-1,b:2,c:3,n:-2,y,x);
  30 y=-sqrt(3)*tanh(sqrt(6)/x-%c)/(sqrt(2)*x);
  31 radcan(ode_check(eq,ans));
  32 0;
  33
  34 /* Case a.i.2 n=2*a, b<0, c<0 */
  35 (print("Special Riccati equation: Case a.i.2: n=2*a, b<0, c<0"),done);
  36 done;
  37 ans:ode1_riccati_special(a:1,b:-1,c:-1,n:2,y,x);
  38 y=-x*tanh(x-%c);
  39 ode_check(eq,ans);
  40 0;
  41 ans:ode1_riccati(eqn:x*'diff(y,x)-y-y^2=-x^2,y,x);
  42 y=-x*tanh(x-%c);
  43 ode_check(eqn,ans);
  44 0;
  45 ans:ode1_riccati_special(a:2,b:-2,c:-3,n:4,y,x);
  46 y=-sqrt(3)*x^2*tanh(sqrt(6)*x^2/2-%c)/sqrt(2);
  47 radcan(ode_check(eq,ans));
  48 0;
  49 ans:ode1_riccati_special(a:3,b:-2,c:-3,n:6,y,x);
  50 y=-sqrt(3)*x^3*tanh(sqrt(6)*x^3/3-%c)/sqrt(2);
  51 radcan(ode_check(eq,ans));
  52 0;
  53 ans:ode1_riccati_special(a:-1,b:-2,c:-3,n:-2,y,x);
  54 y=sqrt(3)*tanh(sqrt(6)/x+%c)/(sqrt(2)*x);
  55 radcan(ode_check(eq,ans));
  56 0;
  57
  58 /* Case a.i.3 n=2*a, b>0, c<0 */
  59 (print("Special Riccati equation: Case a.i.3: n=2*a, b>0, c<0"),done);
  60 done;
  61 ans:ode1_riccati_special(a:1,b:1,c:-1,n:2,y,x);
  62 y=-x*tan(x-%c);
  63 trigsimp(ode_check(eq,ans));
  64 0;
  65 ans:ode1_riccati(eqn:x*'diff(y,x)-y+y^2=-x^2,y,x);
  66 y=-x*tan(x-%c);
  67 trigsimp(ode_check(eqn,ans));
  68 0;
  69 ans:ode1_riccati_special(a:2,b:2,c:-3,n:4,y,x);
  70 y=-sqrt(3)*x^2*tan(sqrt(6)*x^2/2-%c)/sqrt(2);
  71 radcan(trigsimp(ode_check(eq,ans)));
  72 0;
  73 ans:ode1_riccati_special(a:3,b:2,c:-3,n:6,y,x);
  74 y=-sqrt(3)*x^3*tan(sqrt(6)*x^3/3-%c)/sqrt(2);
  75 radcan(ode_check(eq,ans));
  76 0;
  77 ans:ode1_riccati_special(a:-1,b:2,c:-3,n:-2,y,x);
  78 y=sqrt(3)*tan(sqrt(6)/x+%c)/(sqrt(2)*x);
  79 radcan(ode_check(eq,ans));
  80 0;
  81
  82 /* Case a.i.4 n=2*a, b<0, c>0 */
  83 (print("Special Riccati equation: Case a.i.4: n=2*a, b<0, c>0"),done);
  84 done;
  85 ans:ode1_riccati_special(a:1,b:-1,c:1,n:2,y,x);
  86 y=x*tan(x+%c);
  87 ode_check(eq,ans);
  88 0;
  89 ans:ode1_riccati(eqn:x*'diff(y,x)-y-y^2=x^2,y,x);
  90 y=x*tan(x+%c);
  91 ode_check(eqn,ans);
  92 0;
  93 ans:ode1_riccati_special(a:2,b:-2,c:3,n:4,y,x);
  94 y=sqrt(3)*x^2*tan(sqrt(6)*x^2/2+%c)/sqrt(2);
  95 radcan(ode_check(eq,ans));
  96 0;
  97 ans:ode1_riccati_special(a:3,b:-2,c:3,n:6,y,x);
  98 y=sqrt(3)*x^3*tan(sqrt(6)*x^3/3+%c)/sqrt(2);
  99 radcan(ode_check(eq,ans));
 100 0;
 101 ans:ode1_riccati_special(a:-1,b:-2,c:3,n:-2,y,x);
 102 y=-sqrt(3)*tan(sqrt(6)/x-%c)/(sqrt(2)*x);
 103 radcan(ode_check(eq,ans));
 104 0;
 105 kill(a,b,c,n);
 106 done;
 107
 108 /* Check that the sign of b and c is handled correctly */
 109 assume(b>0,c>0);
 110 [b>0,c>0];
 111 ode1_riccati_special(2,b,c,4,y,x);
 112 y=sqrt(c)*x^2*tanh(sqrt(b)*sqrt(c)*x^2/2+%c)/sqrt(b);
 113 ans:ode1_riccati(eqn:x*'diff(y,x)-2*y+b*y^2=c*x^4,y,x);
 114 y=sqrt(c)*x^2*tanh(sqrt(b)*sqrt(c)*x^2/2+%c)/sqrt(b);
 115 ode_check(eqn,ans);
 116 0;
 117 forget(b>0,c>0);
 118 [b>0,c>0];
 119
 120 assume(b<0,c<0);
 121 [b<0,c<0];
 122 ans:ode1_riccati_special(2,b,c,4,y,x);
 123 y=-sqrt(-c)*x^2*tanh(sqrt(-b)*sqrt(-c)*x^2/2-%c)/sqrt(-b);
 124 ans:ode1_riccati(eqn:x*'diff(y,x)-2*y+b*y^2=c*x^4,y,x);
 125 y=-sqrt(-c)*x^2*tanh(sqrt(-b)*sqrt(-c)*x^2/2-%c)/sqrt(-b);
 126 ode_check(eqn,ans);
 127 0;
 128 forget(b<0,c<0);
 129 [b<0,c<0];
 130
 131 assume(b>0,c<0);
 132 [b>0,c<0];
 133 ode1_riccati_special(2,b,c,4,y,x);
 134 y=-sqrt(-c)*x^2*tan(sqrt(b)*sqrt(-c)*x^2/2-%c)/sqrt(b);
 135 ans:ode1_riccati(eqn:x*'diff(y,x)-2*y+b*y^2=c*x^4,y,x);
 136 y=-sqrt(-c)*x^2*tan(sqrt(b)*sqrt(-c)*x^2/2-%c)/sqrt(b);
 137 ode_check(eqn,ans);
 138 0;
 139 forget(b>0,c<0);
 140 [b>0,c<0];
 141
 142 assume(b<0,c>0);
 143 [b<0,c>0];
 144 ode1_riccati_special(2,b,c,4,y,x);
 145 y=sqrt(c)*x^2*tan(sqrt(-b)*sqrt(c)*x^2/2+%c)/sqrt(-b);
 146 ans:ode1_riccati(eqn:x*'diff(y,x)-2*y+b*y^2=c*x^4,y,x);
 147 y=sqrt(c)*x^2*tan(sqrt(-b)*sqrt(c)*x^2/2+%c)/sqrt(-b);
 148 ode_check(eqn,ans);
 149 0;
 150 forget(b<0,c>0);
 151 [b<0,c>0];
 152
 153 /* Tests for ode1_riccati_special - Case ii: (n-2*a)/2n=k , positive integer */
 154 /* k=2 */
 155 ans:ode1_riccati_special(a:1,b:1,c:1,n:-2/3,y,x);
 156 y=1/((1/3-1/(tanh(3/x^(1/3)-%c)*x^(1/3)))*x^(2/3))+1;
 157 ode_check(eq,ans);
 158 0;
 159 ans:ode1_riccati_special(a:2,b:1,c:1,n:-4/3,y,x);
 160 y=1/((2/3-1/(tanh(3/(2*x^(2/3))-%c)*x^(2/3)))*x^(4/3))+2;
 161 ode_check(eq,ans);
 162 0;
 163 ans:ode1_riccati_special(a:-3/2,b:1,c:1,n:1,y,x);
 164 y=x/(sqrt(x)/tanh(2*sqrt(x)+%c)-1/2)-3/2;
 165 ode_check(eq,ans);
 166 0;
 167 ans:ode1_riccati_special(a:3/2,b:1,c:1,n:-1,y,x);
 168 y=1/((1/2-1/(tanh(2/sqrt(x)-%c)*sqrt(x)))*x)+3/2;
 169 ode_check(eq,ans);
 170 0;
 171 ans:ode1_riccati_special(a:-3/2,b:4,c:9,n:1,y,x);
 172 y=x/(2*sqrt(x)/(3*tanh(12*sqrt(x)+%c))-1/18)-3/8;
 173 ode_check(eq,ans);
 174 0;
 175
 176 /* Tests for ode1_riccati_special - Case iii: (n+2*a)/2n=k , positive integer */
 177 /* k=2 */
 178 ans:ode1_riccati_special(a:1,b:1,c:1,n:2/3,y,x);
 179 y=x^(2/3)/(x^(1/3)/tanh(3*x^(1/3)+%c)-1/3);
 180 ode_check(eq,ans);
 181 0;
 182 ans:ode1_riccati_special(a:2,b:1,c:1,n:4/3,y,x);
 183 y=x^(4/3)/(x^(2/3)/tanh(3*x^(2/3)/2+%c)-2/3);
 184 ode_check(eq,ans);
 185 0;
 186 ans:ode1_riccati_special(a:3/2,b:1,c:1,n:1,y,x);
 187 y = x/(sqrt(x)/tanh(2*sqrt(x)+%c)-1/2);
 188 ode_check(eq,ans);
 189 0;
 190 ans:ode1_riccati_special(a:3,b:1,c:1,n:2,y,x);
 191 y = x^2/(x/tanh(x+%c)-1);
 192 ode_check(eq,ans);
 193 0;
 194 ans:ode1_riccati_special(a:3,b:-9,c:4,n:2,y,x);
 195 y=x^2/(3*x/(2*tan(6*x+%c))-1/4);
 196 ode_check(eq,ans);
 197 0;
 198
 199 /* Tests for ode1_riccati_special - Case c: not integrable  a#0 */
 200
 201 /* Bessel function of integer order - easy to check */
 202 ans:ode1_riccati_special(a:1,b:-1,c:1,n:1,y,x);
 203 y=(((bessel_y(2,2*sqrt(x))-bessel_y(0,2*sqrt(x)))*%c
 204     +bessel_j(2,2*sqrt(x))-bessel_j(0,2*sqrt(x)))*sqrt(x)
 205     -bessel_y(1,2*sqrt(x))*%c-bessel_j(1,2*sqrt(x)))
 206  /(2*bessel_y(1,2*sqrt(x))*%c+2*bessel_j(1,2*sqrt(x)));
 207 ode_check(eq,ans);
 208 0;
 209
 210 ans:ode1_riccati_special(a:1,b:-2,c:3,n:1,y,x);
 211 y = (((sqrt(6)*bessel_y(2,2*sqrt(6)*sqrt(x))
 212          -sqrt(6)*bessel_y(0,2*sqrt(6)*sqrt(x)))
 213          *%c
 214          +sqrt(6)*bessel_j(2,2*sqrt(6)*sqrt(x))
 215          -sqrt(6)*bessel_j(0,2*sqrt(6)*sqrt(x)))
 216          *sqrt(x)
 217          -bessel_y(1,2*sqrt(6)*sqrt(x))*%c-bessel_j(1,2*sqrt(6)*sqrt(x)))
 218          /(4*bessel_y(1,2*sqrt(6)*sqrt(x))*%c
 219           +4*bessel_j(1,2*sqrt(6)*sqrt(x)));
 220 ode_check(eq,ans);
 221 0;
 222
 223 assume(x>0);
 224 [x>0];
 225 ans:ode1_riccati_special(a:2,b:-2,c:3,n:2,y,x);
 226 y = (((sqrt(2)*sqrt(3)*bessel_y(2,2*sqrt(3)*x/sqrt(2))
 227       -sqrt(2)*sqrt(3)*bessel_y(0,2*sqrt(3)*x/sqrt(2)))*%c
 228       +sqrt(2)*sqrt(3)*bessel_j(2,2*sqrt(3)*x/sqrt(2))
 229       -sqrt(2)*sqrt(3)*bessel_j(0,2*sqrt(3)*x/sqrt(2)))*x
 230       -2*bessel_y(1,2*sqrt(3)*x/sqrt(2))*%c
 231       -2*bessel_j(1,2*sqrt(3)*x/sqrt(2)))
 232  /(4*bessel_y(1,2*sqrt(3)*x/sqrt(2))*%c+4*bessel_j(1,2*sqrt(3)*x/sqrt(2)));
 233 ode_check(eq,ans);
 234 0;
 235 forget(x>0);
 236 [x>0];
 237
 238 /* Cases that end up in ode1_riccati_original with m=-2 */
 239 ans:ode1_riccati_special(a:1,b:-1,c:1,n:0,y,x);
 240 y = -(((%c-sqrt(3))*sin(sqrt(3)*log(x)/2)+(sqrt(3)*%c+1)*cos(sqrt(3)*log(x)/2))
 241 /(2*%c*sin(sqrt(3)*log(x)/2)+2*cos(sqrt(3)*log(x)/2)));
 242 ode_check(eq,ans);
 243 0;
 244
 245 ans:ode1_riccati_special(a:1,b:1,c:1,n:0,y,x);
 246 y=((sqrt(5)+1)*x^sqrt(5)+(1-sqrt(5))*%c)/(2*x^sqrt(5)+2*%c);
 247 ode_check(eq,ans);
 248 0;
 249
 250 ans:ode1_riccati_special(a:1,b:-1/4,c:1,n:0,y,x);
 251 y=-((2*%c*log(x)+4*%c+2)/(%c*log(x)+1));
 252 ode_check(eq,ans);
 253 0;
 254
 255 /* Cases with a=0 and b*c<0 */
 256 ans:ode1_riccati_special(a:0,b:-1,c:1,n:2,y,x);
 257 y=(bessel_y(1,x)*%c+bessel_j(1,x))*x/(bessel_y(0,x)*%c+bessel_j(0,x));
 258 ode_check(eq,ans);
 259 0;
 260
 261 ans:ode1_riccati_special(a:0,b:-1,c:1,n:1,y,x);
 262 y=(bessel_y(1,2*sqrt(x))*%c+bessel_j(1,2*sqrt(x)))*sqrt(x)/(bessel_y(0,2*sqrt(x))*%c+bessel_j(0,2*sqrt(x)));
 263 ode_check(eq,ans);
 264 0;
 265
 266 /* Cases with a=0 and b*c>0 */
 267 ans:ode1_riccati_special(a:0,b:1,c:1,n:2,y,x);
 268 y=-((bessel_k(1,x)*%c-bessel_i(1,x))*x/(bessel_k(0,x)*%c+bessel_i(0,x)));
 269 ode_check(eq,ans);
 270 0;
 271
 272 ans:ode1_riccati_special(a:0,b:1,c:1,n:1,y,x);
 273 y=-((bessel_k(1,2*sqrt(x))*%c-bessel_i(1,2*sqrt(x)))*sqrt(x)/(bessel_k(0,2*sqrt(x))*%c+bessel_i(0,2*sqrt(x))));
 274 ode_check(eq,ans);
 275 0;
 276
 277 kill(a,b,c,n);
 278 done;
 279
 280 /* some tests for ode1_riccati_original */
 281 eqn:'diff(y,x)+b*y^2=c*x^m;
 282 'diff(y,x)+b*y^2=c*x^m;
 283 /* Case  m*(2*k+1)+4*k=0 and k=0 => m=0
 284    also  m*(2*k-1)+4*k=0 and k=0 => m=0 */
 285 ans:ode1_riccati_original(b:4,c:1,m:0,y,x);
 286 y=tanh(2*x+%c)/2;
 287 ode_check(eqn,ans);
 288 0;
 289 ans:ode1_riccati_original(b:-4,c:1,m:0,y,x);
 290 y=tan(2*x+%c)/2;
 291 ode_check(eqn,ans);
 292 0;
 293 ans:ode1_riccati_original(b:4,c:-1,m:0,y,x);
 294 y=-tan(2*x-%c)/2;
 295 ode_check(eqn,ans);
 296 0;
 297 ans:ode1_riccati_original(b:-4,c:-1,m:0,y,x);
 298 y=-tanh(2*x-%c)/2;
 299 ode_check(eqn,ans);
 300 0;
 301
 302 /* Case m*(2*k+1)+4*k=0, k positive integer */
 303 /* m=-4/3  k=1 */
 304 ans:ode1_riccati_original(b:1,c:1,m:-4/3,y,x);
 305 y=3*tanh(3*x^(1/3)+%c)/(3*x^(2/3)-tanh(3*x^(1/3)+%c)*x^(1/3));
 306 ode_check(eqn,ans);
 307 0;
 308 /* m=-8/5  k=2 */
 309 ans:ode1_riccati_original(b:-1,c:1,m:-8/5,y,x);
 310 y=-((25*x^(1/5)-5*tan(5*x^(1/5)-%c))/(25*tan(5*x^(1/5)-%c)*x+15*x^(4/5)-3*tan(5*x^(1/5)-%c)*x^(3/5)));
 311 /* This is correct - checking takes too long for testsuite */
 312 /* m=-12/7  k=3 */
 313 ans:ode1_riccati_original(b:1,c:1,m:-12/7,y,x);
 314 y=(343*tanh(7*x^(1/7)+%c)*x^(2/7)-147*x^(1/7)+21*tanh(7*x^(1/7)+%c))/(343*x^(8/7)-294*tanh(7*x^(1/7)+%c)*x+105*x^(6/7)-15*tanh(7*x^(1/7)+%c)*x^(5/7));
 315 /* This is correct - checking takes too long for testsuite */
 316
 317 /* Case m*(2*k-1)+4*k=0, k positive integer */
 318 /* m=-4, k=1 */
 319 ans:ode1_riccati_original(b:1,c:1,m:-4,y,x);
 320 y=(x*tanh((%c*x-1)/x)+1)/(x^2*tanh((%c*x-1)/x));
 321 ode_check(eqn,ans);
 322 0;
 323 /* m=-8/3 k=2 */
 324 ans:ode1_riccati_original(b:1,c:1,m:-8/3,y,x);
 325 y=(tanh((%c*x^(1/3)-3)/x^(1/3))*x+3*x^(2/3)+3*tanh((%c*x^(1/3)-3)/x^(1/3))*x^(1/3))/(tanh((%c*x^(1/3)-3)/x^(1/3))*x^2+3*x^(5/3));
 326 ode_check(eqn,ratsimp(exponentialize(ans)));
 327 0;
 328 /* m=-12/5 k=3 */
 329 ans:ode1_riccati_original(b:1,c:1,m:-12/5,y,x);
 330 y=(3*tanh((%c*x^(1/5)-5)/x^(1/5))*x^(3/5)+15*x^(2/5)+30*tanh((%c*x^(1/5)-5)/x^(1/5))*x^(1/5)+25)/(3*tanh((%c*x^(1/5)-5)/x^(1/5))*x^(8/5)+15*x^(7/5)+25*tanh((%c*x^(1/5)-5)/x^(1/5))*x^(6/5));
 331 /* ode_check(eqn,ratsimp(exponentialize(ans)));
 332 0;  eventually */
 333 /* m=-16/7 k=4 */
 334 ans:ode1_riccati_original(b:1,c:1,m:-16/7,y,x);
 335 y=(15*tanh((%c*x^(1/7)-7)/x^(1/7))*x^(4/7)+105*x^(3/7)+315*tanh((%c*x^(1/7)-7)/x^(1/7))*x^(2/7)+490*x^(1/7)+343*tanh((%c*x^(1/7)-7)/x^(1/7)))/(15*tanh((%c*x^(1/7)-7)/x^(1/7))*x^(11/7)+105*x^(10/7)+294*tanh((%c*x^(1/7)-7)/x^(1/7))*x^(9/7)+343*x^(8/7));
 336 /* ode_check(eqn,ratsimp(exponentialize(ans)));
 337 0;  eventually */
 338
 339 /* Bessel function solution - integer order with b*c<0 */
 340 ans:ode1_riccati_original(b:3,c:-2,m:-1,y,x);
 341 y = -((((sqrt(6)*bessel_y(2,2*sqrt(6)*sqrt(x))
 342        -sqrt(6)*bessel_y(0,2*sqrt(6)*sqrt(x)))*%c
 343        +sqrt(6)*bessel_j(2,2*sqrt(6)*sqrt(x))
 344        -sqrt(6)*bessel_j(0,2*sqrt(6)*sqrt(x)))*sqrt(x)
 345     -bessel_y(1,2*sqrt(6)*sqrt(x))*%c-bessel_j(1,2*sqrt(6)*sqrt(x)))
 346   /((6*bessel_y(1,2*sqrt(6)*sqrt(x))*%c+6*bessel_j(1,2*sqrt(6)*sqrt(x)))*x));
 347 ode_check(eqn,ans);
 348 0;
 349
 350 /* Bessel function solution - integer order with b*c>0 */
 351 assume(x>0);
 352 [x>0];
 353 ans:ode1_riccati_original(b:3,c:2,m:-1,y,x);
 354 y = -((((sqrt(6)*bessel_k(2,2*sqrt(6)*sqrt(x))
 355         +sqrt(6)*bessel_k(0,2*sqrt(6)*sqrt(x)))*%c
 356         -sqrt(6)*bessel_i(2,2*sqrt(6)*sqrt(x))
 357         -sqrt(6)*bessel_i(0,2*sqrt(6)*sqrt(x)))*sqrt(x)
 358         -bessel_k(1,2*sqrt(6)*sqrt(x))*%c-bessel_i(1,2*sqrt(6)*sqrt(x)))
 359   /((6*bessel_k(1,2*sqrt(6)*sqrt(x))*%c
 360     +6*bessel_i(1,2*sqrt(6)*sqrt(x)))*x));
 361 ode_check(eqn,ans);
 362 0;
 363 forget(x>0);
 364 [x>0];
 365
 366 /* Bessel function solution - non-integer order b*c<0 */
 367 ans:ode1_riccati_original(b:1,c:-1,m:2,y,x);
 368 y = -((((bessel_j(3/4,x^2/2)-bessel_j(-5/4,x^2/2))*%c
 369        +bessel_j(5/4,x^2/2)-bessel_j(-3/4,x^2/2))*x^2
 370        -bessel_j(-1/4,x^2/2)*%c-bessel_j(1/4,x^2/2))
 371     /((2*bessel_j(-1/4,x^2/2)*%c+2*bessel_j(1/4,x^2/2))*x));
 372 ode_check(eqn,ans);
 373 0;
 374
 375 /* Bessel function solution - non-integer order b*c>0 */
 376 assume(x>0);
 377 [x>0];
 378 ans:ode1_riccati_original(b:2,c:3,m:-5,y,x);
 379 y = -(((-bessel_k(1/3,-(2*sqrt(6))/(3*x^(3/2)))*%c)
 380           -bessel_i(-1/3,-(2*sqrt(6))/(3*x^(3/2))))
 381           *x^2
 382           +((sqrt(6)*bessel_k(4/3,-(2*sqrt(6))/(3*x^(3/2)))
 383            +sqrt(6)*bessel_k(2/3,-(2*sqrt(6))/(3*x^(3/2))))
 384            *%c
 385            -sqrt(6)*bessel_i(2/3,-(2*sqrt(6))/(3*x^(3/2)))
 386            -sqrt(6)*bessel_i(-4/3,-(2*sqrt(6))/(3*x^(3/2))))
 387            *sqrt(x))
 388           /((4*bessel_k(1/3,-(2*sqrt(6))/(3*x^(3/2)))*%c
 389            +4*bessel_i(-1/3,-(2*sqrt(6))/(3*x^(3/2))))
 390            *x^3);
 391 ode_check(eqn,ans);
 392 0;
 393 forget(x>0);
 394 [x>0];
 395
 396 /* Original Riccati equation: m=-2 Case i */
 397 ans:ode1_riccati_original(b:2,c:-3,m:-2,y,x);
 398 y=((%c-sqrt(23))*sin(sqrt(23)*log(x)/2)+(sqrt(23)*%c+1)*cos(sqrt(23)*log(x)/2))
 399   /(4*%c*x*sin(sqrt(23)*log(x)/2)+4*x*cos(sqrt(23)*log(x)/2));
 400 ode_check(eqn,ans);
 401 0;
 402
 403 /* Original Riccati equation: m=-2 Case ii */
 404 ans:ode1_riccati_original(b:2,c:3,m:-2,y,x);
 405 y=(3*x^5-2*%c)/(2*x^6+2*%c*x);
 406 ode_check(eqn,ans);
 407 0;
 408
 409 /* Original Riccati equation: m=-2 Case iii */
 410 ans:ode1_riccati_original(b:1,c:-1/4,m:-2,y,x);
 411 y=(%c*log(x)+2*%c+1)/(2*%c*x*log(x)+2*x);
 412 ode_check(eqn,ans);
 413 0;
 414
 415 kill(b,c,m);
 416 done;
 417
 418 /* Murphy 249 (3-3) */
 419 ans:contrib_ode(eqn:3*x*'diff(y,x)=3*x^(2/3)+(1-3*y)*y,y,x);
 420 [y=tanh(3*x^(1/3)+%c)*x^(1/3)];
 421 ode_check(eqn,ans[1]);
 422 0;
 423
 424 /* murphy 373  (3-2) */
 425 assume(a>0);
 426 [a>0];
 427 ans:ode1_riccati(eqn:x^4*'diff(y,x)+a^2+x^4*y^2=0,y,x);
 428 y=(x*tan((%c*x-a)/x)+a)/(x^2*tan((%c*x-a)/x));
 429 ode_check(eqn,ans);
 430 0;
 431 forget(a>0);
 432 [a>0];
 433
 434 /* Murphy 168 - Case (a.iii) with k = 2 */
 435 ans:ode1_riccati(eqn:x*'diff(y,x)-y+y^2=x^(2/3),y,x);
 436 y=x^(2/3)/(x^(1/3)/tanh(3*x^(1/3)+%c)-1/3);
 437 ode_check(eqn,ans);
 438 0;