Question
The solution of the differential equation with
exists, but is not unique.
Answer :
Word Count : 179
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Consider the differential equation (\frac{dy}{dx} = y) with the initial condition (y(0) = 0). The equation is separable. Writing it as (\frac{dy}{dx} = y), we have [ \frac{dy}{y} = dx, \quad y \neq 0. ] Integrating both sides gives [ \ln|y| = x + C \implies y = Ce^x, \quad C \text{ constant}. ] Applying
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Related Questions
Solve: .
Solve the differential equation: .
State whether the following statements are true or false. Justify your answer with the help of a short proof or a counter example:
a) y2 is an integrating factor of the differential equation:
.
) Solve the equation .
Solve: .
बताइए कि निम्नलिखित कथन सत्य हैं या असत्य। संक्षिप्त उपपत्ति अथवा प्रत्युदाहरण की सहायता से अपने उत्तर की पुष्टि कीजिए :
क) y2 अवकल समीकरण :$
का समाकलन गुणक है।
ख) अवकल समीकरण , जहाँ
, के हल का अस्तित्व है लेकिन हल अद्वितीय नहीं है।
ग) अंतराल में समीकरण :
समघात रैखिक समीकरण है।
घ) अवकल समीकरण$
के हल का अस्तित्व है, परंतु हल अद्वितीय नहीं है।
ङ) फैफियन अवकल समीकरण समाकलनीय है।
) हल कीजिए :