IGNOU PHE 5 2026 - निम्नलिखित साधारण अवकल समीकरणों को हल करें : (i) (ii

Question

निम्नलिखित साधारण अवकल समीकरणों को हल करें :

(i) $x dx + y dy + 4y^3(x^2 + y^2) dy = 0$

(ii) $y'' + 4y = 2 \cos x \cos 3x$

29 Jan 2026

Answer :

Word Count : 531

(i) समीकरण है: [ x , dx + y , dy + 4y^3(x^2 + y^2) , dy = 0 ] इसे पहले व्यवस्थित करें: [ x , dx + y(1 + 4y^2(x^2 + y^2)) , dy = 0 ] या फिर: [ x , dx + y , dy + 4y^3(x^2 + y^2) , dy = 0 ] ध्यान दें कि (y , dy + 4y^3(x^2 + y^2) , dy = y(1 + 4y^2(x^2 + y^2)) , dy) इसे अलग करें: [ x , dx + y(1 + 4y^2(x^2 + y^2)) , dy = 0 ] हम देखते हैं कि यह समीकरण ((x^2 + y^2 + y^4(x^2 + y^2))' = ?) के रूप में लिखा जा सकता है। मान लें (r^2 = x^2 + y^2)। तब (dr^2 = 2x dx + 2y dy \implies x dx + y dy = \frac{1}{2} dr^2)। समीकरण बनता है: [ \frac{1}{2} dr^2 + 4y^3 r^2 dy = 0 \implies dr^2 + 8 y^3 r^2 dy = 0 ] यहाँ (r^2 = x^2 + y^2), तो समीकरण बनता है: [ \frac{dr^2}{r^2} = -8 y^3 dy ] दोनों तरफ इंटीग्रेट करें: [ \int \frac{dr^2}{r^2} = \int -8y^3 dy ] [ \ln |r^2| = -2y^4 + C ] [ \ln |x^2 + y^2| + 2y^4 = C ] अतः

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