Class 12 Differential Equations · Very Long Answer5 Markshomogeneous_differential_equations

Class 12 Maths · Differential Equations · Very Long Answer #73

Show that [x sin²(y/x) − y]dx + xdy = 0 is homogeneous and solve it with y(1) = π/4.

✅ Step-by-Step Solution

Final Answer: cot(y/x) = 1 + ln x; choose the continuous branch y/x ∈ (0, π), x > 0.

Rearrange to y′ = y/x − sin²(y/x), a function only of y/x.
Put v = y/x. Then xv′ = −sin²v.
Near v = π/4, integrate csc²v dv = −dx/x to obtain −cot v = −ln x + C.
The initial condition gives cot(π/4) = 1, hence cot v = 1 + ln x.
Thus cot(y/x) = 1 + ln x for x > 0, on the continuous branch y/x ∈ (0, π) through π/4.

[1 mark] Rearrange to y′ = y/x − sin²(y/x), a function only of y/x.
[1 mark] Put v = y/x. Then xv′ = −sin²v.
[1 mark] Near v = π/4, integrate csc²v dv = −dx/x to obtain −cot v = −ln x + C.
[1 mark] The initial condition gives cot(π/4) = 1, hence cot v = 1 + ln x.
[1 mark] Thus cot(y/x) = 1 + ln x for x > 0, on the continuous branch y/x ∈ (0, π) through π/4.

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