Out of the following systems with input x(t) and output y(t), identify the linear system.
Answer & step-by-step solution
A system is linear when input, output and output derivatives appear only to the first power and are not multiplied together; coefficients may still depend on time. In option B, dy/dt+2t y=t²x satisfies this condition. The other listed equations contain squared signal terms. Therefore option B is linear.
Concept and Reasoning
A system is linear when input, output and output derivatives appear only to the first power and are not multiplied together; coefficients may still depend on time. In option B, dy/dt+2t y=t²x satisfies this condition. The other listed equations contain squared signal terms. Therefore option B is linear.
Calculation / Key Relation
Linear form: a1(t)dy/dt+a0(t)y=b0(t)x
Exam Takeaway
Time-varying coefficients do not make a system nonlinear.
Common Mistake
Look for powers/products of x, y or derivatives, not merely the presence of t-dependent coefficients.
Quick Trick
Time-varying coefficients do not make a system nonlinear.
इनपुट x(t) और आउटपुट y(t) वाली निम्नलिखित प्रणालियों में से रैखिक प्रणाली पहचानिए।
उत्तर और चरणबद्ध हल
B — dy/dt + 2t·y(t) = t²x(t)। प्रणाली रैखिक है जब इनपुट, आउटपुट और उनके अवकलज केवल प्रथम घात में हों और आपस में गुणा न हों; गुणांक समय पर निर्भर हो सकते हैं। अन्य विकल्पों में वर्ग पद हैं।
विस्तृत हल (मूल भाषा में)
Concept and Reasoning
A system is linear when input, output and output derivatives appear only to the first power and are not multiplied together; coefficients may still depend on time. In option B, dy/dt+2t y=t²x satisfies this condition. The other listed equations contain squared signal terms. Therefore option B is linear.
Calculation / Key Relation
Linear form: a1(t)dy/dt+a0(t)y=b0(t)x
Exam Takeaway
Time-varying coefficients do not make a system nonlinear.
Common Mistake
Look for powers/products of x, y or derivatives, not merely the presence of t-dependent coefficients.
Quick Trick
Time-varying coefficients do not make a system nonlinear.
ઇનપુટ x(t) અને આઉટપુટ y(t) વાળી નીચેની પ્રણાલીઓમાંથી રેખીય પ્રણાલી ઓળખો.
જવાબ અને પગલાવાર ઉકેલ
B — dy/dt + 2t·y(t) = t²x(t). પ્રણાલી રેખીય છે જ્યારે ઇનપુટ, આઉટપુટ અને તેમનાં વિકલજ ફક્ત પ્રથમ ઘાતમાં હોય અને પરસ્પર ગુણાકાર ન હોય; ગુણાંકો સમય પર આધાર રાખી શકે છે. બાકીના વિકલ્પોમાં વર્ગ પદ છે.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Concept and Reasoning
A system is linear when input, output and output derivatives appear only to the first power and are not multiplied together; coefficients may still depend on time. In option B, dy/dt+2t y=t²x satisfies this condition. The other listed equations contain squared signal terms. Therefore option B is linear.
Calculation / Key Relation
Linear form: a1(t)dy/dt+a0(t)y=b0(t)x
Exam Takeaway
Time-varying coefficients do not make a system nonlinear.
Common Mistake
Look for powers/products of x, y or derivatives, not merely the presence of t-dependent coefficients.
Quick Trick
Time-varying coefficients do not make a system nonlinear.
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