3.9.2 Problems 101 to 200

Table 3.509: First order ode linear in derivative

#

ODE

Mathematica

Maple

101

\[ {}6 y+x y^{\prime } = 3 x y^{\frac {4}{3}} \]

102

\[ {}y^{3} {\mathrm e}^{-2 x}+2 x y^{\prime } = 2 x y \]

103

\[ {}\sqrt {x^{4}+1}\, y^{2} \left (x y^{\prime }+y\right ) = x \]

104

\[ {}y^{3}+3 y^{2} y^{\prime } = {\mathrm e}^{-x} \]

105

\[ {}3 y^{2} y^{\prime } x = 3 x^{4}+y^{3} \]

106

\[ {}{\mathrm e}^{y} x y^{\prime } = 2 \,{\mathrm e}^{y}+2 \,{\mathrm e}^{2 x} x^{3} \]

107

\[ {}2 x \cos \left (y\right ) \sin \left (y\right ) y^{\prime } = 4 x^{2}+\sin \left (y\right )^{2} \]

108

\[ {}\left ({\mathrm e}^{y}+x \right ) y^{\prime } = -1+x \,{\mathrm e}^{-y} \]

109

\[ {}2 x +3 y+\left (3 x +2 y\right ) y^{\prime } = 0 \]

110

\[ {}4 x -y+\left (-x +6 y\right ) y^{\prime } = 0 \]

111

\[ {}3 x^{2}+2 y^{2}+\left (4 x y+6 y^{2}\right ) y^{\prime } = 0 \]

112

\[ {}3 x^{2}+2 x y^{2}+\left (2 x^{2} y+4 y^{3}\right ) y^{\prime } = 0 \]

113

\[ {}x^{3}+\frac {y}{x}+\left (\ln \left (x \right )+y^{2}\right ) y^{\prime } = 0 \]

114

\[ {}1+{\mathrm e}^{x y} y+\left ({\mathrm e}^{x y} x +2 y\right ) y^{\prime } = 0 \]

115

\[ {}\cos \left (x \right )+\ln \left (y\right )+\left ({\mathrm e}^{y}+\frac {x}{y}\right ) y^{\prime } = 0 \]

116

\[ {}x +\arctan \left (y\right )+\frac {\left (x +y\right ) y^{\prime }}{1+y^{2}} = 0 \]

117

\[ {}3 x^{2} y^{3}+y^{4}+\left (3 x^{3} y^{2}+4 x y^{3}+y^{4}\right ) y^{\prime } = 0 \]

118

\[ {}{\mathrm e}^{x} \sin \left (y\right )+\tan \left (y\right )+\left ({\mathrm e}^{x} \cos \left (y\right )+x \sec \left (y\right )^{2}\right ) y^{\prime } = 0 \]

119

\[ {}\frac {2 x}{y}-\frac {3 y^{2}}{x^{4}}+\left (-\frac {x^{2}}{y^{2}}+\frac {1}{\sqrt {y}}+\frac {2 y}{x^{3}}\right ) y^{\prime } = 0 \]

120

\[ {}\frac {2 x^{\frac {5}{2}}-3 y^{\frac {5}{3}}}{2 x^{\frac {5}{2}} y^{\frac {2}{3}}}+\frac {\left (-2 x^{\frac {5}{2}}+3 y^{\frac {5}{3}}\right ) y^{\prime }}{3 x^{\frac {3}{2}} y^{\frac {5}{3}}} = 0 \]

121

\[ {}x^{3}+3 y-x y^{\prime } = 0 \]

122

\[ {}3 y^{2}+x y^{2}-x^{2} y^{\prime } = 0 \]

123

\[ {}x y+y^{2}-x^{2} y^{\prime } = 0 \]

124

\[ {}{\mathrm e}^{x}+2 x y^{3}+\left (\sin \left (y\right )+3 y^{2} x^{2}\right ) y^{\prime } = 0 \]

125

\[ {}3 y+x^{4} y^{\prime } = 2 x y \]

126

\[ {}2 x y^{2}+x^{2} y^{\prime } = y^{2} \]

127

\[ {}2 x^{2} y+x^{3} y^{\prime } = 1 \]

128

\[ {}2 x y+x^{2} y^{\prime } = y^{2} \]

129

\[ {}2 y+x y^{\prime } = 6 x^{2} \sqrt {y} \]

130

\[ {}y^{\prime } = 1+x^{2}+y^{2}+y^{2} x^{2} \]

131

\[ {}x^{2} y^{\prime } = x y+3 y^{2} \]

132

\[ {}6 x y^{3}+2 y^{4}+\left (9 y^{2} x^{2}+8 x y^{3}\right ) y^{\prime } = 0 \]

133

\[ {}y^{\prime } = 1+x^{2}+y^{2}+x^{2} y^{4} \]

134

\[ {}x^{3} y^{\prime } = x^{2} y-y^{3} \]

135

\[ {}3 y+y^{\prime } = 3 x^{2} {\mathrm e}^{-3 x} \]

136

\[ {}y^{\prime } = x^{2}-2 x y+y^{2} \]

137

\[ {}{\mathrm e}^{x}+{\mathrm e}^{x y} y+\left ({\mathrm e}^{y}+{\mathrm e}^{x y} x \right ) y^{\prime } = 0 \]

138

\[ {}2 x^{2} y-x^{3} y^{\prime } = y^{3} \]

139

\[ {}3 x^{5} y^{2}+x^{3} y^{\prime } = 2 y^{2} \]

140

\[ {}3 y+x y^{\prime } = \frac {3}{x^{\frac {3}{2}}} \]

141

\[ {}\left (-1+x \right ) y+\left (x^{2}-1\right ) y^{\prime } = 1 \]

142

\[ {}x y^{\prime } = 12 x^{4} y^{\frac {2}{3}}+6 y \]

143

\[ {}{\mathrm e}^{y}+\cos \left (x \right ) y+\left ({\mathrm e}^{y} x +\sin \left (x \right )\right ) y^{\prime } = 0 \]

144

\[ {}9 y^{2} x^{2}+x^{\frac {3}{2}} y^{\prime } = y^{2} \]

145

\[ {}2 y+\left (1+x \right ) y^{\prime } = 3+3 x \]

146

\[ {}9 \sqrt {x}\, y^{\frac {4}{3}}-12 x^{\frac {1}{5}} y^{\frac {3}{2}}+\left (8 x^{\frac {3}{2}} y^{\frac {1}{3}}-15 x^{\frac {6}{5}} \sqrt {y}\right ) y^{\prime } = 0 \]

147

\[ {}3 y+x^{3} y^{4}+3 x y^{\prime } = 0 \]

148

\[ {}x y^{\prime }+y = 2 \,{\mathrm e}^{2 x} \]

149

\[ {}y+\left (2 x +1\right ) y^{\prime } = \left (2 x +1\right )^{\frac {3}{2}} \]

150

\[ {}y^{\prime } = 3 x^{2} \left (7+y\right ) \]

151

\[ {}y^{\prime } = 3 x^{2} \left (7+y\right ) \]

152

\[ {}y^{\prime } = -x y+x y^{3} \]

153

\[ {}y^{\prime } = \frac {-3 x^{2}-2 y^{2}}{4 x y} \]

154

\[ {}y^{\prime } = \frac {3 y+x}{-3 x +y} \]

155

\[ {}y^{\prime } = \frac {2 x +2 x y}{x^{2}+1} \]

156

\[ {}y^{\prime } = \cot \left (x \right ) \left (\sqrt {y}-y\right ) \]

415

\[ {}y^{\prime } = 1+y^{2} \]

448

\[ {}3 y+y^{\prime } = {\mathrm e}^{-2 t}+t \]

449

\[ {}-2 y+y^{\prime } = {\mathrm e}^{2 t} t^{2} \]

450

\[ {}y+y^{\prime } = 1+t \,{\mathrm e}^{-t} \]

451

\[ {}\frac {y}{t}+y^{\prime } = 3 \cos \left (2 t \right ) \]

452

\[ {}-2 y+y^{\prime } = 3 \,{\mathrm e}^{t} \]

453

\[ {}2 y+t y^{\prime } = \sin \left (t \right ) \]

454

\[ {}2 t y+y^{\prime } = 2 t \,{\mathrm e}^{-t^{2}} \]

455

\[ {}4 t y+\left (t^{2}+1\right ) y^{\prime } = \frac {1}{\left (t^{2}+1\right )^{2}} \]

456

\[ {}y+2 y^{\prime } = 3 t \]

457

\[ {}-y+t y^{\prime } = t^{2} {\mathrm e}^{-t} \]

458

\[ {}y+y^{\prime } = 5 \sin \left (2 t \right ) \]

459

\[ {}y+2 y^{\prime } = 3 t^{2} \]

460

\[ {}-y+y^{\prime } = 2 \,{\mathrm e}^{2 t} t \]

461

\[ {}2 y+y^{\prime } = t \,{\mathrm e}^{-2 t} \]

462

\[ {}2 y+t y^{\prime } = t^{2}-t +1 \]

463

\[ {}\frac {2 y}{t}+y^{\prime } = \frac {\cos \left (t \right )}{t^{2}} \]

464

\[ {}-2 y+y^{\prime } = {\mathrm e}^{2 t} \]

465

\[ {}2 y+t y^{\prime } = \sin \left (t \right ) \]

466

\[ {}4 t^{2} y+t^{3} y^{\prime } = {\mathrm e}^{-t} \]

467

\[ {}\left (t +1\right ) y+t y^{\prime } = t \]

468

\[ {}-\frac {y}{2}+y^{\prime } = 2 \cos \left (t \right ) \]

469

\[ {}-y+2 y^{\prime } = {\mathrm e}^{\frac {t}{3}} \]

470

\[ {}-2 y+3 y^{\prime } = {\mathrm e}^{-\frac {\pi t}{2}} \]

471

\[ {}\left (t +1\right ) y+t y^{\prime } = 2 t \,{\mathrm e}^{-t} \]

472

\[ {}2 y+t y^{\prime } = \frac {\sin \left (t \right )}{t} \]

473

\[ {}\cos \left (t \right ) y+\sin \left (t \right ) y^{\prime } = {\mathrm e}^{t} \]

474

\[ {}\frac {y}{2}+y^{\prime } = 2 \cos \left (t \right ) \]

475

\[ {}\frac {2 y}{3}+y^{\prime } = -\frac {t}{2}+1 \]

476

\[ {}\frac {y}{4}+y^{\prime } = 3+2 \cos \left (2 t \right ) \]

477

\[ {}-y+y^{\prime } = 1+3 \sin \left (t \right ) \]

478

\[ {}-\frac {3 y}{2}+y^{\prime } = 2 \,{\mathrm e}^{t}+3 t \]

479

\[ {}y^{\prime } = \frac {x^{2}}{y} \]

480

\[ {}y^{\prime } = \frac {x^{2}}{\left (x^{3}+1\right ) y} \]

481

\[ {}\sin \left (x \right ) y^{2}+y^{\prime } = 0 \]

482

\[ {}y^{\prime } = \frac {3 x^{2}-1}{3+2 y} \]

483

\[ {}y^{\prime } = \cos \left (x \right )^{2} \cos \left (2 y\right )^{2} \]

484

\[ {}x y^{\prime } = \sqrt {1-y^{2}} \]

485

\[ {}y^{\prime } = \frac {-{\mathrm e}^{-x}+x}{{\mathrm e}^{y}+x} \]

486

\[ {}y^{\prime } = \frac {x^{2}}{1+y^{2}} \]

487

\[ {}y^{\prime } = \left (-2 x +1\right ) y^{2} \]

488

\[ {}y^{\prime } = \frac {-2 x +1}{y} \]

489

\[ {}x +y y^{\prime } {\mathrm e}^{-x} = 0 \]

490

\[ {}r^{\prime } = \frac {r^{2}}{x} \]