Show that the following differential equation is homogeneous and

Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Advertisement

Differential Equations

Long Answer Type

Advertisement

201. Show that the following differential equation is homogeneous and find a primitive of it. Derive the solution wherever possible:
$1 over straight x cos straight y over straight x dx minus open parentheses straight x over straight y sin straight y over straight x plus cos straight y over straight x close parentheses dy space equals 0$

The given differential equation is

straight y over straight x cos straight y over straight x dx minus open parentheses straight x over straight y sin straight y over straight x plus cos straight y over straight x close parentheses dy space equals space 0

or

dy over dx space equals space fraction numerator begin display style straight y over straight x end style cos begin display style straight y over straight x end style over denominator begin display style straight x over straight y end style sin begin display style straight y over straight x end style plus cos begin display style straight y over straight x end style end fraction

Put

straight y space equals space space straight v space straight x space so space that space dy over dx space equals space straight v plus straight x dv over dx

therefore space space space space space space space space space space straight v plus straight x dv over dx space equals space fraction numerator straight v space cos space straight v over denominator begin display style 1 over straight v end style sinv space plus cos space straight v end fraction space space space rightwards double arrow space space space space straight v plus straight x dv over dx space equals fraction numerator straight v squared space cosv over denominator sin space straight v plus straight v space cos space straight v end fraction therefore space space space space space space space straight x dv over dx space equals space fraction numerator straight v squared space cos space straight v over denominator sin space straight v plus straight v space cos space straight v end fraction minus straight v therefore space space space space space straight x dv over dx space equals space fraction numerator straight v squared space cos space straight v minus space straight v space sin space straight v space minus space straight v squared space cos space straight v over denominator sin space straight v plus straight v space cos space straight v end fraction therefore space space space space straight x dv over dx space equals negative fraction numerator straight v space sin space straight v over denominator sin space straight v plus straight v space cos space straight v end fraction

Separating the variables and integrating ,we get,

integral fraction numerator sin space straight v plus space straight v space cosv over denominator straight v space sinv end fraction dv space equals negative integral 1 over straight x dx

therefore space space log space open vertical bar straight v space sinv close vertical bar space equals space minus space log space open vertical bar straight x close vertical bar space plus space log space open vertical bar straight A close vertical bar rightwards double arrow space space space log space open vertical bar straight y over straight x sin space straight y over straight x close vertical bar plus space log space open vertical bar straight x close vertical bar space equals space log space open vertical bar straight A close vertical bar rightwards double arrow space space log space open vertical bar straight y space sin straight y over straight x close vertical bar space minus space log space open vertical bar straight x close vertical bar space plus space log space open vertical bar straight x close vertical bar space equals space log space open vertical bar straight A close vertical bar rightwards double arrow space space log space open vertical bar straight y space sin straight y over straight x close vertical bar space equals space log space open vertical bar straight A close vertical bar rightwards double arrow space space space space space space straight y space sin space straight y over straight x space equals space straight A comma space space space which space is space required space solution. space

74 Views

Advertisement

202. Show that the following differential equation is homogeneous and find a primitive of it. Derive the solution wherever possible:

2 space straight y space straight e to the power of straight x over 4 end exponent space dx plus open parentheses straight y minus 2 space straight x space straight e to the power of straight x over straight y end exponent close parentheses space dy space equals space 0

73 Views

203. Show that the following differential equation is homogeneous and find a primitive of it. Derive the solution wherever possible:

straight y space dx space plus space straight x space open parentheses log space straight y over straight x close parentheses space dy space minus space 2 space straight x space dy space equals space 0

74 Views

204. Show that the given differential equation is homogeneous and solve it:

left parenthesis straight x squared plus xy right parenthesis space dy space equals space left parenthesis straight x squared plus straight y squared right parenthesis space dx

75 Views

Advertisement

Short Answer Type

205. Show that the given differential equation is homogeneous and solve it:

straight y apostrophe space equals space fraction numerator straight x plus straight y over denominator straight x end fraction

73 Views

206. Show that the given differential equation is homogeneous and solve it:
(x-y) dy - (x+y) dx = 0

74 Views

207. Show that the given differential equation is homogeneous and solve it:

open parentheses straight x squared minus straight y squared close parentheses space dx space plus space 2 xy space dy space equals space 0

80 Views

Long Answer Type

208. Show that the given differential equation is homogeneous and solve it:

straight x squared dy over dx space equals space straight x squared minus 2 straight y squared plus straight x space straight y

73 Views

Advertisement

209. Show that the given differential equation is homogeneous and solve it:

open curly brackets straight x space cos space open parentheses straight y over straight x close parentheses plus straight y space sin space open parentheses straight y over straight x close parentheses close curly brackets straight y space dx space equals space open curly brackets straight y space sin space open parentheses straight y over straight x close parentheses minus straight x space cos space open parentheses straight y over straight x close parentheses close curly brackets space straight x space dy

76 Views

210. Solve

straight x space dy space minus space straight y space dx space equals space square root of straight x squared plus straight y squared end root space dx.

79 Views

Advertisement