JEE Mains · Maths · STD 12 - 9. differential equations
The solution of the differential equation \(x\,\frac{{dy}}{{dx}}\, + \,2y\, = \,{x^2}\,(x\, \ne \,0)\) with \(y(1) = 1,\) is
- A \(y\, = \,\frac{{{x^3}}}{5}\, + \,\frac{1}{{5{x^2}}}\)
- B \(y\, = \,\frac{{{x^2}}}{4}\, + \,\frac{3}{{4{x^2}}}\)
- C \(y\, = \,\frac{4}{5}{x^3}\, + \,\frac{1}{{5{x^2}}}\)
- D \(y\, = \,\frac{3}{4}{x^2}\, + \,\frac{1}{{4{x^2}}}\)
Answer & Solution
Correct Answer
(B) \(y\, = \,\frac{{{x^2}}}{4}\, + \,\frac{3}{{4{x^2}}}\)
Step-by-step Solution
Detailed explanation
\(x \frac{d y}{d x}+2 y=x^{2}: y(1)=1\) \(\frac{d y}{d x}+\left(\frac{2}{x}\right) y=x(L D E \text { in } y)\) \({\rm{IF}} = {{\rm{e}}^{\int {\frac{2}{{\rm{x}}}{\rm{dx}}} }} = {{\rm{e}}^{2\ln {\rm{x}}}} = {{\rm{x}}^2}\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The sum of all the roots of the equation \( (x-1)^{2}-5|x-1|+6=0 \), is:JEE Mains 2026 Easy
- If \(\mathrm{y}=\mathrm{y}(\mathrm{x})\) is an implicit function of \(\mathrm{x}\) such that \(\log _{e}(x+y)=4 x y\), then \(\frac{d^{2} y}{d x^{2}}\) at \(x=0\) is equal to .... .JEE Mains 2021 Hard
- Let \(y=x\) be the equation of a chord of the circle \(C_{1}\) (in the closed half-plane \(x\ge0)\) of diameter 10 passing through the origin. Let \(C_{2}\) be another circle described on the given chord as its diameter. If the equation of the chord of the circle \(C_{2}\), which passes through the point (2, 3) and is farthest from the center of \(C_{2}\), is \(x+ay+b=0,\) then \(a-b\) is equal to:JEE Mains 2026 Easy
- Let \(\alpha, \beta(\alpha>\beta)\) be the roots of the quadratic equation \(x ^{2}- x -4=0\). If \(P _{ a }=\alpha^{ n }-\beta^{ n }, n \in N\), then \(\frac{ P _{15} P _{16}- P _{14} P _{16}- P _{15}^{2}+ P _{14} P _{15}}{ P _{13} P _{14}}\) is equal to\(......\)JEE Mains 2022 Hard
- Let \(f\) : \(A \to B\) be a function defined as \(f(x)\, = \frac{{x - 1}}{{x - 2}}\) , where \(A\, = R - \{2\}\) and \(B\, = R - \{1\}\) . Then \(f\) isJEE Mains 2018 Hard
- If the tangents drawn to the hyperbola \(4y^2 = x^2 + 1\) intersect the co-ordinate axes at the distinct points \(A\) and \(B\), then the locus of the mid point of \(AB\) isJEE Mains 2018 Hard
More PYQs from JEE Mains
- The area (in sq. units) of the region \(\left\{(\mathrm{x}, \mathrm{y}) \in \mathrm{R}^{2} | 4 \mathrm{x}^{2} \leq \mathrm{y} \leq 8 \mathrm{x}+12\right)\) isJEE Mains 2020 Hard
- If the line \(y\, = \,mx\, + \,7\sqrt 3 \) is normal to the hyperbola \(\frac{{{x^2}}}{{24}} - \frac{{{y^2}}}{{18}} = 1,\) then a value of \(m\) isJEE Mains 2019 Hard
- The relation \(R=\{(x, y): x, y \in \mathbb{Z}\) and \(x+y\) is even \(\}\) is:JEE Mains 2025 Medium
- For the curve \(C :\) \(\left(x^{2}+y^{2}-3\right)+\left(x^{2}-y^{2}-1\right)^{5}=0\), the value of \(3 y^{\prime}-y^{3} y^{\prime \prime}\), at the point \((\alpha, \alpha), \alpha>0\), on \(C\), is equal to.JEE Mains 2022 Hard
- Let \(I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x\). If \(I(0)=3\), then \(\mathrm{I}\left(\frac{\pi}{12}\right)\) is equal to :JEE Mains 2024 Hard
- Let \(z_1, z_2 \in \mathbb{C}\) be the distinct solutions of the equation \(z^2 + 4z - (1 + 12i) = 0\). Then \(|z_1|^2 + |z_2|^2\) is equal to :JEE Mains 2026 Medium