Verified Solution • 2025–26 Session moderate

QJEEM24Maf049: JEE Main 2024 Mathematics - Class 12 JEEM24Maf049 Solution: If , then is equal to __________.

If dx dy=1+x−y2y,x(1)=1\frac{\mathrm{d} x}{\mathrm{~d} y}=\frac{1+x-y^2}{y}, x(1)=1 , then 5x(2)5 x(2) is equal to __________.

Step-by-Step Verified Solution

dxdy−xy=1−y2y\frac{d x}{d y}-\frac{x}{y}=\frac{1-y^2}{y}

Integrating factor =e∫−1ydy=1y=\mathrm{e}^{\int-\frac{1}{y} d y}=\frac{1}{y}

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padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-top: 0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" tabindex="0" ctxtmenu_counter="7" style="font-size: 129.3%; position: relative;"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;"><mjx-mtable style="min-width: 10.312em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-bottom: 0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-top: 0.15em; 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padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-top: 0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd></mtd><mtd><mi>x</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mn>1</mn><mo>=</mo><mo>−</mo><mn>1</mn><mo>−</mo><mn>1</mn><mo>+</mo><mi>c</mi><mo stretchy="false">⇒</mo><mi>c</mi><mo>=</mo><mn>3</mn></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn><mo>−</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>y</mi></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mn>5</mn><mi>x</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo>−</mo><mn>4</mn><mo>+</mo><mn>6</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mn>5</mn></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container>

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